0 < y < 1$$

Let $Z = XY^2$ and $W = Y$ be a joint transformation of (X,Y)

Sketch the graph of the support of $(Z,W)$ and describe it

mathematically.

I'm not very sure how to describe (Z,W).

First, I draw the graph of the support of X and Y, which is a rectangular support.

Now I "map" each interval over to (Z,W).

For $x=0, 0<y<1, z=0, 0<w<1$

For $x=1, 0<y<1, 0<z<w^2, 0<w<1 $

For $y=0, 0<x<1, z=0, w=0$

For $y=1...

Finding support for multivariate transformation]]>

it comes to finding the CDF using the distribution function technique.

I know that support of y is 0 ≤y<4, and it is

not a one-to-one transformation.

Now, I am confused with part b), finding the limits when calculating the cdf of Y.

Here's my working.

When -1<x<1, it's a two-to-one transformation, 0≤ y<1

P(Y≤y) = P(X^2≤y)

= P(

When -2<x<-1, it's a one-to-one transformation, 1< y<4

P(Y≤y) =...

Integral limits when using distribution function technique]]>

I want to know what is the incresing and decreasing interval of this even function $|e^x+e^{-x}|?$

If any member knows the correct answer, may reply to this question.]]>

Let X be the time taken for the cars to arrive.

Given that 1 car passes every 2 mins, theta = 2.

We are interested in the 10th car that passes so alpha = 10.

Thus I know the distribution is Gamma~(theta=2,alpha=10)

I can solve this using the gamma distribution to find P(X>28.41). (Although the summation is from k= 0 to 9 which is pretty tedious)

To relate the chi-squared in this case, r = 20 while theta = 2, but I don't...

Chi-squared link to gamma]]>

I understand that for X>3, Y=6-X and for X<3, Y=X.

For X = 3, Y=3

For part b, I got P(Y>y)= (3-y)/3, for 0≤y<3

Now for part c, I know P(Y>y) relates to the cdf.

But the definition of cdf relates to P(Y<y), so I'm guessing I have to

do 1-P(Y>y) to get the cdf which is y/3, 0≤y<3.

I'm thinking the pdf would be 1/3 for 0≤y<3?

I know for sure Y is a uniform distribution.

I'm not too sure on the interval (x-a)/(b-a)

Is it Y~Uniform...

Uniform distribution question]]>

Internet users per 100 data, Award winners per 10 million...

Determine whether there is sufficient evidence to support a claim of linear correlation]]>

Find the expected salary of workers at skill level $Z=z$ had they received $x$ years of college education. [Hint: Use Theorem 4.3.2, with $e:Z=z,$ and the...

Counterfactual Expectation Calculation]]>

Here's why you should study causality: because once you've...

The Causal Revolution and Why You Should Study It]]>

I am trying to solve this problem:

r balls are randomly assigned into n urns. The assignment is random and the balls are cannot be distinguished. What is the probability that exactly m urns will contain exactly k balls each ?

I know that the probability of each ball to be in each urn is 1/n.

I addition I have (nCm) ways to choose which are the urns to be filled with k balls.

Then I have r−km balls to distribute in the n−m remaining urns.

I do not know how to proceed. Can...

r balls in n urns]]>

Assume that there is an $\varepsilon>0$ such that whenever $\mu_0$ and $\nu_o$ are point distributions on $S$ (in other words, $\mu_0$ and $\nu_0$ are Direac masses) we have

$$\|\mu_0P-\nu_0P\|_{TV}\leq \varepsilon$$

Now let $Y=Y_0, Y_1, Y_2, \ldots$ be an independent copy of $X$.

Regarding mixing time of a Markov chain]]>

possible edges on the vertex set ${1, . . . , n}$, and for each $i$, let $A_i$ be the event that $e_i ∈ E(G)$.

Prove that the events $A_1, . . . , A_{n \choose 2}$ are independent.

Can I just have some help understanding what details I should be including here?

It's so trivial that I don't know how to write it down as a proof.

There is a probability of $\frac{1}{2}$...

Choosing Edges, Random Graph]]>

$Hessian = \frac{\partial^2 f}{\partial x_i \partial x_j}$

However, I have come across a different expression, source: https://users.ugent.be/~yrosseel/lavaan/lavaan2.pdf (slide 40)

$nCov(\hat\theta) =A^{-1}=[-Hessian]^{-1} = [-\partial F(\hat\theta)/(\partial\hat\theta\partial\hat\theta')]^{-1}$

A - represents a hessian matrix.

I am curious are...

Unfamiliar hessian matrix expression]]>

I've been trying to solve this exercise for a long time. I know that is not that dificult.

However i cant remember how to do it.

Thanks in advance.

Calculate the F-test statistic for this equation and use it to perform a test for the null hypothesis that the slope coefficient is equal to zero

Ho = 0

H1 not equal to 0

1977 observations.

I found this relation between the R-squared and the F-test. Where does it come...

F-Test probabilty]]>

In an effort to evaluate the quality of a new fertilizer we selected 12 fields which we split exactly in the middle. In the one half we used the old fertilizer and in the other half we used the new fertilizer. Then we measured in tonnes the crop that was harvested from each piece of land.

The results are in the following table:

- Should we use the dependent or the independent samples method to compare crops from the old and...

Is the new fertilizer more efficient than the old one?]]>

The below table shows the average monthly gross income of a sample of 44 developers. For each individual sample, it is indicated their country of employment and years of service in their field.

Calculate the regression line with the dependent variable the monthly gross income and independent the years of employee service and check the significance of the F criterion at $\alpha = 0.05$.

I have done the following:

Therefore we get...

F-Test : Do we accept the null-hypothesis?]]>

View attachment estimation-min.jpg]]>

In a study at $15$ children at the age of $10$ years the number of hours of television watching per week and the pounds above or below the ideal body weight were determined (high positive values = overweight).

- Determine the simple linear regression equation by considering the weights above the ideal body weight as a dependent variable.
- Perform a significance test for the slope of the regression line at significance level $\alpha = 5\%$ (using p-values).
- ...

Perform a significance test]]>

We have data of a sample of $100$ people from a population with standard deviation $\sigma=20$.

We consider the following test: \begin{align*}H_0 : \ \mu\leq 100 \\ H_1 : \ \mu>100\end{align*}

The real mean is $\mu=102$ and the significance level is $\alpha=0.1$.

I want to calculate the probability of the error of type 2.

I have done the following:

The statistic function is...

Probability of the error of type 2]]>

In the following table there are the heights of the employees of a company:

- Calculate the mean value, the variance and the standard deviation of the heights of the employees.
- Determine the distribution of sampling with replacement of average number of children of each employee for sample size $2$.
- Which is the mean value and the variance of the sampling average?
- Which is the mean value of the sampling variances...

Heights of the employees of a company]]>

We are given a list of $300$ data which are the square meters of houses. I have calculated the mean value and the median. After that we have to say something about the symmetry of the distribution. For that do we have to make a diagram from the given data? Is there a program to do that? ]]>

Let $X_n$ be independent random variables such that $X_1 = 1$, and for $n \geq 2$,

$P(X_n=n)=n^{-2}$ and $P(X_n=1)=P(X_n=0)=\frac{1}{2}(1-n^{-2})$.

Show $(1/\sqrt{n})(\sum_{m=1}^{n}X_n-n/2)$ converges weakly to a normal distribution as $n \rightarrow \infty$.

My professor sent this problem over email and I am first off wondering about the notation. I think that the last line is meant to read $(\frac{1}{\sqrt{n}})(\frac{\sum_{m=1}^{n}X_m-n}{2})$? (I...

Weak Convergence to Normal Distribution]]>

$P(\displaystyle\bigcap_{i=1}^n A_i)=\displaystyle\sum_{i=1}^n P(A_i)-\displaystyle\sum_{i<j} P(A_i\cup A_j)+\displaystyle\sum_{i<j<k} P(A_i\cup A_j\cup A_k)-\cdots - (-1)^n P(A_1\cup A_2\cup ... \cup A_n).$

Hello, the Hint is use induction on $n$.]]>

Suppose that Carl wants to estimate the proportion of books that he likes, denoted by 𝜃. He modeled

𝜃 as a probability distribution given in the following table. In the year 2019, he likes 17 books out of a

total of 20 books that he read. Using this information, determine 𝜃̂ using Maximum a Posteriori method.

_____________

𝜃 | 0.8 | 0.9 |

𝑝(𝜃 )| 0.6 |0.4 |

_____________

My attempt at a solution:

I know I have to use Bayes theorem to solve this, so the equation...

Maximum a posteriori]]>

I start from level 0. There is a probability p chance to drop to level -1 and a (1-p) chance to increase to level 1.

The levels range from level -n to level n. When it reaches level -n or level n, it resets back to 0 on the same cycle.

(Also, if you are at level -1, there is (1-p) chance to go back to level 0)

How do I calculate the percentage of landing on each level (not counting the reset)...

Need help with probability question. probability of dependent events.]]>

Sorry, in my first message, I posted this question in the Basic Probability section, and so I moved it to this section.

I have a surface (for example, a blank paper).

In this surface, I have some elements of the set "A" randomly distributed.

In this surface, I also have some elements of the set "B" randomly distributed.

I would like to understand how may elements of "B" are present within a ray X from any element of "A".

I mean something like: "for each element An, there are N%...

Compute probability closeness between points in a 2D surface]]>

X1[k] is a Bernoulli random variable with P=0.5 and

X2[k] is a Bernoulli random variable with P=0.7 for all k>=0

Let Y be a random process formed by merging X1 and X2, i.e. Y[k] =1 if and only if X1[k] = X2[k] = 1 and Y[k] = 0 otherwise.

a.) Solve for the success probability of Y if X1 and X2 are uncorrelated.

b.) Solve for the success probability of Y if E[X1[k]X2[k]] = 0.3.

c.) If E[X1[k]X2[k]] is constant for all k, find the minimum...

Merged Bernoulli process]]>

First post here. I have some data I am trying to do some forecasting on and was hoping somebody who knows what they're actually doing can verify what I have done.

A few years ago, the company I work for developed a mobile app for its customers and about 1 year ago they added some new features. The CTO came to me and asked me "Can you please give me a 12 month estimate on the number of customers using our mobile app?" and the data I have access to is:

(1) The number of customers...

Forecasting metric using regression. Is this a sound approach?]]>

1. The domain $S$ is partitioned into $i \in 1,...,n$ subdomains.

2. Each subdomain is searched one after another, with each search lasting for the same length of time $T/t$ for some positive integer t. These subdomain can be...

How to optimise the following probability / algorithm / combinatorics problem?]]>

1. Suppose that the service time for a student enlisting during enrollment is modeled as an

exponential RV with a mean time of 1 minute. If the school expects 500 students during

enrollment period, what is the probability that the registration committee will finish enlisting all

students within a day if a day constitutes a total of 8 working hours? Use the Central Limit

Theorem.

2. Suppose that you have a class at 8:45 am and you...

Poisson process]]>

Joy and Ethan have agreed to meet for dinner between 8:00 PM and 9:00 PM. Suppose that Ethan may

arrive at any time between the set meeting. Joy on the other hand will arrive at the set meeting under the

following conditions:

• Joy will always arrive earlier than Ethan.

• Joy will never arrive later than 20 minutes.

• Joy’s arrival time added to Ethan’s arrival time will never exceed an hour.

Let X be the arrival time (in minutes) of Ethan and Y be...

Joint CDF of a word problem]]>

1. What is the value of b such that f(x) becomes a valid density function

2. What is the cumulative distribution function F(x) of f(x)

3. What is the Expectation of X, E[X]

4. What is the Variance of X, Var[X]

5. What is the probability that X is within one standard deviation from the mean

So far, I've gotten b by integrating the function from 1.5 to 4 and setting it equal to 1...

cdf, expectation, and variance of a random continuous variable]]>

Here is my (imaginary) problem :

I have to make a medical diagnosis for a patient : is he A or B ?

I can not make inspection of the patient, and my only help is a poll of N medical experts that observed him.

Out of N experts, the track records for their past diagnoses are different:

- 22% of experts had a success rate of 40%

- 41% of experts had a succes rate of 53%

- 37% of experts had a success rate of 68%

Questions :

1 / How can I maximise my...

Maximize chance of success based on opinion poll]]>

A discrete random variable

I want to express the probability function of

How can I answer this question?

If any member knows the correct answer, he/she may reply with correct answer.]]>

I need your help with this probability problem.

I have random variable x

The vector of random variables X

Now we do an experiment. In each experiment we cast x

Computing success percentage]]>

answer the following questions

- Formulate null and alternative hypotheses. What do you think...

correlational analysis]]>

Basically, I attempt to roll 5 times, and I have 70% chance for the roll to even happen. When roll happens I have 1/14 chance of rolling the right result. So...

I roll 5 times, with 30% chance of failing to roll and 1/14 chance of the roll succeeding.]]>

a) compute the critical value C* such that for C<= C* the prcesss B will die out with probability one and for C > C* the process will survuve with positive probability.

b) compute the survival probability p1 that the process B starting with one individual will not die out as a function of C .

c) compute the survival probability...

A Galton Watson branching process]]>

At present, I am studying CDF, PDF and MGF techniques for transformations of random...

Distribution and Density functions of maximum of random variables]]>

[Tex] Z=1+X+XY^2[/Tex]

[Tex]W=1+X[/Tex]

I want to find

[Tex]Cov(Z,W)=Cov(1+X+XY^2,1+X)[/Tex]

[Tex]Cov(Z,W)=Cov(X+XY^2,X)[/Tex]

[Tex]Cov(Z,W)=Cov(X,X)+Cov(XY^2,X)[/Tex]

[Tex]Cov(Z,W)=Var(X)+E(X^2Y^2)-E(XY^2)E(X)[/Tex]

[Tex]Cov(Z,W)=1+E(X^2)E(Y^2)-E(X)^2E(Y^2)[/Tex]

[Tex]Cov(Z,W)=1+1-0=2[/Tex]

Now E(X)=0, So [Tex]E(X)^2E(Y^2)=0[/Tex], But i don't follow how...

E(x^2)if X is N(0,1).]]>

My maths problem involves probability relating to lotteries.

The odds of winning a lottery jackpot are 1 in 14 million. Over a period of time 50 million tickets have been bought, so the expected number of jackpot winners would be 3.5. However the actual number of jackpot winners is 7.

Is it possible to calculate a p value to prove that although the actual number of winners is higher than expected, it is still within normal expected ranges? Or are there any other statistical models...

Actual vs Expected Number of Lottery Winners]]>

higher floor chosen at random. What is the expected number of stops the elevator makes? Generalize the

result for a k -store building with k>2: Estimate the value k0 such that the probability that the elevator

will stop exactly 10 times is larger equal than 0.9 for all k>k0.

I've done this so far not sure if it is correct

the Expectation of number of stops = 7 . (1-(6/7)^10)= 5.5]]>

Specific model purchases = B

Ratio of specific options = %

A = 225,000

B= 9560

Option 1 = 41% of all B

Option 2 = 9% of all A

Option 3 = 6% of all A

Option 4 = 65% of all A

Option 5 = 30% of A

Option 6 = 9% of all B

Probability or rarity of one vehicle having all these attributes?]]>

I swear ... I seem to struggle trying to decipher which ANOVA to use ... drives me crazy.

In this study, I have 75 participants that are all varsity athletes. They all completed a survey that scored attentional style within 6 variables (BET, OET, BIT, OIT, NAR, RED).

I am looking to see if there is any difference between the participants scores and the sports they play as well as the position they play in that sport. I was also considering to see if there is a difference...

Looking for Appropriate Test to Use]]>

I am looking the following concering boolean algebra.

For a certain test group, it is found that $42\%$ of the people have never skied yet, that $58\%$ of them have never flown yet, and that $29\%$ of them have already flown and skied.

Which probability is higher then:

To meet someone, who has already skied, from the group of those who have never flown, or to find someone who has already flown from the group of those who have already skied?

I have done the following:

Let...

Which probability is higher?]]>

I am considering using an independent t-test for two samples. Is this the correct approach...

Need assistance to decide with analysis to use for problem]]>

walking velocity (cm/s)

cadence (steps/min)

stride length (cm)

What would be the appropriate graph to run with RStudio to account for the three different measurements?

Thanks,

Dants]]>

I would like to illustrate the interpretation of any sex-related differences in grip strength that might i fI consider the amount of musculature in the...

Require assistance with possible ANOVA use]]>

My goal is to facilitate the simple estimation of percentage body fat. I wish to generate a predictive equation the best estimate of...

Require assistance with possible multiple regression analysis]]>

Participants performed three jump trials in each of the following conditions: ankle mobility restricted and a control...

Requesting assistance for possible ANCOVA analysis]]>

First off, assume that state 1 is Chinese and that state 2 is Greek, and state 3 is Italian.

A student never eats the same kind of food for 2 consecutive weeks. If she eats a Chinese restaurant one week, then she is equally likely to have Greek as Italian food the next week. If she eats a Greek restaurant one week, then she...

Markov chains- Can I have some help creating the transition matrix for this scenario?]]>

A wheel of fortune is divided into $10$ equal sized parts, of which one of them brings the jackpot.

- A player wants to investigate the regularity of the wheel of fortune. He turns the wheel 20 times. Calculate the probability that he will get at least one main prize if it is a laplace wheel.
- How often does the player have to spin the wheel of fortune to get at least one main prize with a probability of at least 90%.
- The player objects to the wheel when less...

A wheel of fortune is divided into 10 parts, one of them brings the jackpot.]]>

Tim rolls $20$ times a fair dice.

- Calculate the probability to get a "$3$" exactly $8$ times.
- The probability to get at least $k$ times a "$5$" is equal to $10,18\%$ Calculate $k$.
- Explain the relation to the term $\displaystyle{\sum_{i=0}^2B\left (10;\frac{5}{6};i\right )}$.

I have done the following:

- The probability to get a "$3$" is equal to $\frac{1}{6}$. If we throw $20$ times the dice the probability to get exactly eight "$3$"...

Rolling 20 times a fair dice]]>

Now how to find the values of $a$ such that $\{Y_n\}_{n\geq 1}$ is martingale, submartingale, supermartigale?

Solution:I have no idea to answer this question. I searched on internet, but i didn't get any clew about the way using which one can answer this question till now. If...

Finding out the sequence as Martingale]]>

Is the estimate \(\displaystyle \frac{1}{n} \sum_{i}^{n} f(Xi)\) always unbiased for \(\displaystyle \frac{1}{N} \sum_{i}^{N} f(xi)\) no matter what f is?

My thinking: I don't think all f's are unbiased, because not all sample parameters (ex: variance, or s^2) are unbiased for the population parameter (unless they are corrected for finite population sampling). I am confused if I am interpreting the...

Bias of functions defined on samples for population]]>

Book Recommendation, Please: Time Series Analysis]]>

The problem:

$X,Y,Z$ are random variables that are dependent and uniformly-distributed in $[0,1]$, and let $\alpha$ be a given number in $[0,1]$. I am asked to compute the following:

$\text{Pr}(X+Y+Z>\alpha \;\;\; \& \;\;\; X+Y\leq \alpha)$

What I have so far

$f_{X+Y+Z,X+Y}(u,v)=f_{Z,X+Y}(u-v,v)=f_{Z}(u-v)\cdot f_{X+Y}(v)$

(1) Is the above equation correct? I think it stands for discrete RVs but not quite sure for continuous RVs... If...

Joint cumulative distribution of dependent variables]]>

example number 101 can assume 0 to 100.

my goal is to choose 24 of this 78,000 prime numbers and predict if this 24 are odd or even

Who Have you a solutions?

I could have a history of extraction but i don't know if could be a help.

Thanks]]>

Hint given by author:- Find their joint moment generating functions.

Answer: Now Joint MGf of $X+Y={e^{\mu}}^{2t}+\sigma^2t^2$ and of $X-Y=1$. So, joint MGF of $X+Y+X-Y$ is $e^{2\mu t}+ \sigma^2 t^2$. This indicates they are independent. Is their any other method in advanced calculus?]]>

Thank you much for any constructive comment.]]>

I have the following question in one of my tutorials. I need some help in resolving this.

Background: A manufacturing company developed 40000 new drugs and they need to be tested.

Question

The QA checks on the previous batches of drugs found that — it is 4 times more likely that a drug is able to produce a better result than not.

If we take a sample of ten drugs, we need to find the theoretical probability that at most three drugs are not able to do a satisfactory job.

a.)...

Calculating theoretical probability]]>

Consider a sequence of Bernoulli trials with success probability $p$. Fix a positive integer $r$ and let $\mathcal{E}$ denote the event that a run of $r$ successes is observed; recall that we do not allow overlapping runs. We use a recurrence relation for $u_n$, the probability that $\mathcal{E}$ occurs on the $n$th trial, to derive the generating function $U(s)$. Consider $n \ge r$ and the event that trials $n, n − 1, . . . , n − r + 1$ all...

Success runs in Bernoulli trials]]>

A wind turbine manufacturer would like to increase the throughput of its production system. For this purpose it intends to install a buffer between the pre-assembly and the final assembly of the wind turbines. The manufacturer can generate a profit of 10.000 Euro per wind turbine. However, buffer spaces are also fairly expensive. The company estimates that one buffer space costs 5000 Euro per month. The...

Markov Chain Problem]]>

I've been using a really simplistic equation for this and I'd like to get an actual table/equation/and standard deviations if possible.

Here are the rules.

roll xd6s (six-sided dice)

every die that rolls a 4 or greater is a "hit."

6s rolled count as a hit and then "explode," adding another die which is then rolled. That die hits on a 4+ and also explodes on 6s (this explosion chain can continue infinitely, theoretically).

What is the average number of hits for any value of x...

Average "Hits" on exploding dice]]>

The random variables $X_1, X_2, \ldots , X_{10}$ are independent and have the same distribution function and each of them gets exactly the values $\pm 2$ and with equal probability.

We define the random variable $S=X_1+X_2+\ldots +X_{10}$.

I want to calculate $\mathbb{E}(S^2)$.

Could you give me a hint how we could calculate that? I don't really have an idea. ]]>

A couple gets $n$ children. At each birth, the probability to get a boy is $p$ (independent births). Which is the probability that exactly $k$ of the children are boys?

I have thought the following:

Let $X$ be the number of boys that the couple gets. Then the desired probality is

$P(X=k)=p^k \cdot (1-p)^{n-k}$

Am I right? ]]>

A picture in which pixel either takes 1 with a prob of q and 0 with a prob of 1-q, where q is the realized value of a r.v Q which is uniformly distributed in interval [0,1]

Let Xi be the value of pixel i, we observe for each pixel the value Yi = Xi + N where N is normal with mean 2 and unit variance.( Same noise distrib everywhere) Assume that conditional on Q the Xi's are independent...

Sum of Random Variables..]]>

Data are collected on the wingspans of adult robins. For N=20 birds, the sample mean and variance are given by \(\displaystyle \overline{x}\)=9.5cm and \(\displaystyle s^{2}=2.6^{2}cm^{2}\)

a) If we assume that the true population variance, \(\displaystyle \sigma^{2}\), is known to be \(\displaystyle 2.6^{2}cm^{2}\) (i.e. using a Z-test), construct a 95% confidence interval for the population...

Confidence Interval, p-value and Critical Region]]>

The random variables and are described by a joint PDF which is uniform on the triangular set defined by the constraints 0 <= x <= 1, 0<= theta <= x Find the LMS estimate of theta given that X = x , for in the range [0,1] . Express your answer in terms x.

I started by calculating the joint pdf by first calculating the area of the triangle which is 1/2 * x * 1 = x /2 . The joint pdf will be 1 / (x/2) = 2 / x

Then I...

Least Mean Square Question]]>

thanks for help]]>

Intuitively, for $E\in \mathcal X$, we think of $P_x(E)$ as the probability of...

"Two step" Markov chain is also a Markov chain.]]>

n indipendent trials are carried out.

We want to know the probability density function of the random variable Y, that is defined as the average value of the “n” outcomes of the trials described above.]]>

2 coins are thrown 20 times. I want to calculate the probability

(a) to achieve exactly 5 times the Tail/Tail

(b) to achieve at least 2 times Tail/Tail

If we throw the 2 coins once the probability that we get Tail/Tail is equal to $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$, or not?

Is the probability then at (a) equal to $\left (\frac{1}{4}\right )^{20}$ ?

Could you give me a hint for (b) ?

]]>

We have the density function $f_x(x)=\frac{2c^2}{x^3}, x\geq 0, c\geq 0$.

I want to calculate the maximum Likelihood estimator for $c$.

We have the Likelihood Function $$L(c)=\prod_{i=1}^nf_{X_i}(x_i;c)=\prod_{i=1}^n\frac{2c^2}{x_i^3}$$

The logarithm of the Likelihood function is \begin{align*}\ell (c)&=\ln L(c)=\ln \left (\prod_{i=1}^n\frac{2c^2}{x_i^3}\right )=\sum_{i=1}^n\ln \frac{2c^2}{x_i^3}=\sum_{i=1}^n \left [\ln (2c^2)-\ln (x_i^3)\right ]\\ &=n\ln (2c^2)-\sum_{i=1}^n...

Maximum Likelihood estimator]]>

A friend of mine on another forum, knowing I am involved in the math help community, approached me regarding a question in statistics. Here's what he said:

Heteroscedasticity]]>

The edges of each hexagon have been colored with one of three colors randomly.

If you pick two hexagons randomly without replacement, what is the probability that they are the same? (Rotation is okay).

The total space or denominator is 3^(2×6), therefore we have 3^(2×6) at the bottom, but what is the numerator?

The answer is 4263, any ideas?]]>

I have another procedure:

$P(E)=\prod_{i \in I} p_i^{k_i}(1-p_i)^{1-k_i}$

which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find \(\displaystyle P(E)\)?]]>

Also, what's the difference between ANOVA AND T-TEST?]]>

Answer the following Questions:

Hypotheses - Formulate null and alternative hypotheses. What do you think is the relationship between IQ scores and GPA?

Variables – Describe the...

Correlational Analysis]]>

Suppose that $X$ has the uniform distribution on the interval $[0,2]$ and $Y$ has the uniform distribution on the interval $[2,4]$. If $X,Y$ are independent, I want to find the probability that the difference $Y-X$ is $\leq 1$.

I have thought the following.

The density function of $X$ is

$$p_1(x)=\left\{\begin{matrix}

\frac{1}{2} &, 0 \leq x \leq 2 \\

0 & , \text{ otherwise}

\end{matrix}\right.$$

while the density function of $Y$ is

$$p_2(x)=\left\{\begin{matrix}...

Calculate probability]]>

We have that $X$ and $Y$ follow the normal distribution $N(0,1)$ and are independent.

- Which is the distribution of $(X,Y)$ ?
- Which is the covariance table of $(X,Y)$ ?

Is $(X,Y)$ related to the linear combination of $X$ and $Y$ ? ]]>

Y = N + T + F + NT + NF + NTF + error

Y= Grams of seed

N= Number of fruit

T= Type of fruit (2 types, alpha)

F= Field number (3)

I have tried putting this in MiniTab and I cant get this set up correctly.

Assistant> Regression> Multiple Regression

Y= Grams of Seed

Continuous X Variable= Number of Fruit, Field Number - but I can't select Type since they are words and not numbers...Regression Model in MiniTab]]>

A and B play 100 games of squash; A wins E times. B claims that both of them have the same probability of winning a game.

We consider that the games are independent.

(a) Formulate the null hypothesis and the alternative hypothesis.

(b) For which values of $E$ will the null hypothesis be rejected with $\alpha=1\%$ and for which with $\alpha=5\%$ ?

Is at (a) the null hypothesis $H_0: \ p=\frac{1}{2}$ and the alterinative hypothesis $H_1: \ p\neq \frac{1}{2} $ ? Or do we call...

Hypothesis testing - Winning a game]]>

A teacher wants to find out if the order of the exam tasks has an impact on the performance of the students. Therefore, he creates two versions ($ X $ and $ Y $) of an exam in which the exam tasks are arranged differently. The versions are randomly distributed so that $ n $ students receive version $ X $, and $ m = n $ receive version $ Y $ from them. We call the expected score at $ X $ with $ \mu_X $, and the expected score at $ Y $ with $ \mu_Y $. The variances are denoted $...

Performance of students - Hypothesis testing]]>

We have the following hypotheses: $$H_0: \mu\geq 60 \ \ \ , \ \ \ H_1:\mu<60$$

A test is executed with a sample of size $25$ and an estimated standard deviation $S'=8$.

From the test we get a p-value of 5%. I want to determine the value of the mean of the sample $\overline{X}$.

The p-value is equal to $P(T\leq t\mid H_0)$. So is this in this case equal to

$P(\mu\leq 60\mid H_0)$ ?

If yes, then we have the following: $$p=0.05\Rightarrow \Phi \left...

Which is the mean value of the sample?]]>

The variable $ Y $ denotes the amount of money that an adult person gives out for Christmas presents.

The distribution of $ Y $ depends on whether the person is employed ($ E = 1 $) or not ($ E = 0 $).

It holds that $ P (E = 1) = p $, i.e a randomly selected person is employed with probability $ p $.

We have the following

\begin{align*}&E(Y\mid E=1)=\mu_1 \\ &V(Y\mid E=1)=\sigma_1^2 \\ &E(Y\mid E=0)=\mu_0 \\ &V(Y\mid E=0)=\sigma_0^2 \\ &E(Y)=\mu=p\mu_1+(1-p)\mu_0 \\...

Expected values and variances]]>

I know that I need to use law of total probability and define A

a) How much people read only one newspaper?

b) How much people read at least two newspapers?

c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?

d) How much people read one morning newspaper and one evening...

3 newspapers- a question about Inclusion–exclusion principle]]>

A public transport company claims that its buses are at least $95\%$ on time. (A bus is still on time here, if he has at most $3$ minutes delay compared to the timetable.) A sample size of $n = 1000$ at various stops results in $66$ delays. The probability that a randomly selected bus will arrive on time is denoted by $p$.

(a) You as a passenger doubt the claim of the enterprise. Test the company's claim to a significance level of $\alpha = 0, 10$.

(b) Use an example to explain...

Hypothesis test - delay]]>

Throw a cube until you get the number 6, then stop throwing.

a) What is the sample space of the experiment?

b) Let's call the event to throw the cube n times En. How much elements from the sample space are within E

**The cube is a standard six-sided die, with the numbers "1" thru "6" printed on the sides**]]>

A student answers a question in American test that has m options that are given as follows:

In probability P the student has learned the question and therefore knows how to choose the correct answer, otherwise he guesses the question.

a) What is the probability that the student studied the subject of the question given that he answered correct on the question?

b) Analyze the result for m=1 and m->inf]]>

A research institute wants to establish a confidence interval for the quota of working people in a city. Let $\hat{p}_n $ be the estimated quota, based on a sample of size $n$. It is assumed that $n> 30$.

How can one determine the length of the confidence interval?

Generally this is equal to $L = 2 \cdot \frac{\sigma \cdot z}{\sqrt{n}}$, right?

Do you have to use the $\hat{p}_n$ in the formula in this case? But how?

]]>

The variable $X$ is normally distributed with unknown expected value $\mu$ and unknown variance $\sigma^2$.

I want to determine the confidence interval for $\mu$ for $n=22; \ \overline{X}_n=7.2; \ S'=4; \ 1-\alpha=0.90$.

Is $S'$ the standard deviation? In some notes that I found in Google, it symbolized $s_n$ the standrard deviation of a sample and by $s_n'$ if we have the total population instead of a sample. If it is meant here like that, we follow the same steps...

Confidence interval for μ]]>

We have a playlist with $2000$ songs. The length of the songs on the playlist are on average $3.5$ minutes (i.e. $3$ minutes and $30$ seconds) with a standard deviation of $1.7$ minutes.

1) Can we find the probability that a randomly chosen song is longer than $4.5$ minutes?

2) Can we findthe probability that a random selection of $100$ songs lasts on average at least $4$ minutes?

3) Can we find the probability that a random selection of $200$ songs lasts in total at...

Probabilities about length of songs]]>

The daily turnover $X$ of Cafes has the expected value $ \mu_X = 600 $ Euro and the standard deviation $ \sigma_X = 30 $ Euro.

(a) How many cafes at least have to be surveyed in a random sample, so that $\overline{X}_n$ deviates from $\mu_X$ with a probability of at least $95\%$ by less than $12$ euros?

(b) After a survey of $500$ Cafes the arithmetic mean is $690$. Is this result surprising after the question (a) ?

I have done the following:

(a) From Chebyshev's...

Determine the size of sample]]>

I thought that it could be relationed with the independent status of the machines, because if the machine 1 is working or blocked the machine 2 will be working, blocking or idle, and machine 3 may be working or idle too. That is my only approach about the issue

Right now i have not any further aproximations...

how can i set this problem as a continuous markov chain?]]>

A fair coin is tossed $ n $ times; $ X_i = 1 $ denotes the event that "head" appears in the $ i $-th toss.

a) How are the single toss $X_i$, $=1, \ldots , n$, distributed?

b) How many toss are needed so that the proportion of "head" $\overline{X}_n$ is in the interval $0, 45 < \overline{X}_n < 0, 55$ with probability $90\%$ ?

c) Given that the coin is tossed $100$ times, determine the probability that we get totally at most $30$ times "head".

I have done the following...

Probability that we get totally at most 30 times "head"]]>

Let $ Y=1-X^2 $, where $ X~ U(0,1) $. What statement is TRUE?

-$ E(Y^2)=2 $

- $ E(Y^2)=1/2 $

- $ var(Y)=1/12 $

- $ E(Y)=E(Y^2) $

-None of the remaining statements.

Solution:

I compute: $ E(Y^2)=E(1-X^2)^2=E(1+X^4-2X^2)=1+E(X^4)-2E(X^2) $, then?]]>

The geometric distribution with parameter $p\in (0,1)$ has the probability function \begin{equation*}f_X(x)=p(1-p)^{x-1}, \ \ x=1, 2, 3, \ldots\end{equation*}

I have shown that $f_X$ for each value of $p\in (0,1)$ is strictly monotone decreasing, as follows:

\begin{align*}f_X(x+1)=p(1-p)^{x+1-1}=p(1-p)^{(x-1)+1}=p(1-p)^{x-1}(1-p)\overset{(\star)}{<}p(1-p)^{x-1}=f_X(x)\end{align*} $(\star)$ : Since $p\in (0,1)$ we have that \begin{equation*}0<p<1\Rightarrow -1<-p<0 \Rightarrow...

Greatest probability - Expected value]]>