What is Cdf: Definition and 113 Discussions

In statistics, cumulative distribution function (CDF)-based nonparametric confidence intervals are a general class of confidence intervals around statistical functionals of a distribution. To calculate these confidence intervals, all that is required is an
independently and identically distributed (iid) sample from the distribution and known bounds on the support of the distribution. The latter requirement simply means that all the nonzero probability mass of the distribution must be contained in some known interval



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  1. G

    Finding CDF of a Continuous R.V.

    Homework Statement Let f(x) = (1 + cx)/2 for x between -1 and 1 and f(x)=0 otherwise, where c is between -1 and 1. Show that f is a density and find the corresponding cdf. Find the quartiles and the median of the distribution in terms c. Homework Equations NA The Attempt at a Solution I...
  2. A

    Cdf of a discrete random variable and convergence of distributions

    In the page that I attached, it says "...while at the continuity points x of F_x (i.e., x \not= 0), lim F_{X_n}(x) = F_X(x)." But we know that the graph of F_X(x) is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of...
  3. A

    Continuous Variable pdf from a cdf

    Homework Statement f X,Y(x,y) = (8 +xy^3)/64, if -1<x<1, -2<y<2 0, otherwise Find the probability density function of W = 2X+Y. Homework Equations F(w) = Pr{W≤w}=∫∫f(x,y)dxdy f(w) = d/dw F(w) The Attempt at a Solution I found the support of W to be -4<w<4 I...
  4. S

    How to Determine the Density and Distribution for Transformed Variables Y and Z?

    Homework Statement Density of f_x (x) = 4x^4 for 0<x<1 Y=(x-1/4)^2 Z= X^-2 Determine density of Y and Distribution of Z Homework Equations The cdf of f_x (x) is invalid since F_x (x) = (4/5)x^5 so the limit to infinity does not equal 1 as a cdf should have. Am I missing...
  5. S

    Statistics Problem: finding a PDF using the CDF technique

    Hey guys, I'm stuck on a question in my homework assignment and I was wondering if you could push me in the right direction? So here's the question: X and Y are continuous random variables with joint pdf f(x,y)= 4xy (0<x<1, 0<y<1, and otherwise 0). Find the pdf of T=X+Y using the CDF...
  6. C

    How to get the CDF from a histogram

    Hello, I have a histogram, where I count the number of occurrences that a function takes particular values in the range 0.8 and 2.2. I would like to get the cumulative distribution function for the set of values. Is it correct to just count the total number of occurrences until each...
  7. P

    Statistics proof regarding integration of cdf

    Homework Statement For any cdf F(x) of a continuous random variable, show that \int_{-\infty}^{\infty}[F(x+b)-F(x+a)]dx=b-a for any a<b. Homework Equations The Attempt at a Solution Not really sure where to begin. I figure I can split the integrals and do u subs, and (after...
  8. E

    CDF of X: Probability Pr{X < b}

    Hello, Suppose X is a Chi-square random variable. Then what is: \text{Pr}\left\{X<b\right\}? Does the above probability is the CDF of X? The only difference is that there is no equality! Thanks
  9. H

    MHB V: Understanding the Calculation of Variance for a Given CDF

    I already have the full solution to this so I'm not looking for an answer, I am hoping to get an explanation of certain parts that are unclear. Question: A random variable X has the following CDF: f(x) =\left\{ \begin{array}{lr} 0& x<0\\ \frac{x^2-2x+2}{2} & 1 \le x < 2 \\ 1 & x \ge 2...
  10. E

    The CDF of sum of i.ni.d exponential RVs

    Hello, I need to find the CDF of \mathcal{X}=\sum_{l=0}^L|h(l)|^2 where h(l) is complex Gaussian with zeros mean and variance \sigma^2_l In particular, I need to proof that: \text{Pr}\left[\mathcal{X}\leq b\right]\doteq b^{L+1} where dotted equal means in asymptotic...
  11. A

    Expectation value for CDF fails

    Hi, (Sorry for the slight misnomer in the title... I can't edit it!) I'm doing several problems to compute the expectation value and variances of sub-samples & operations on the normal distribution; and I am having trouble getting results that agree with numerical simulations. I have several...
  12. E

    CDF and PDF of order statistics

    Hi, I have K i.i.d. exponentially distributed random variables with mean unity. I need to find the CDF and PDF of the summation of the largest two random variables. How can I do that? The problem in this case is that the combinations are not independent. Thanks in advance
  13. A

    Binomial Distribution: Find p, given CDF

    I have a question about binomial distribution There is a random var X follows Binomial distribution ~B(n,p), where n is known but p is UNKNOWN. It is also known that a for known value of x, CDF(x) = Pr(X<=x) = 0.9 Is there anyway to estimate p? To give a concrete example, if n=8...
  14. J

    CDF of a variable with a negative exponent in its PDF

    Ok, this one's got me stumped! Let's take as an example the probability density function for a random variable X so that: f(x) = \frac{4}{3x^{3}} 1≤x<2 f(x) = \frac{x}{12} 2≤x≤4 f(x) = 0 So the CDF for this variable comes out as: F(x) = \frac{-2}{3x^{2}} 1≤x<2 F(x) =...
  15. C

    How Do You Calculate the Sum Involving Series and Normal CDF?

    I am trying to calculate the following sum: vt\Phi(a)+\frac{(vt)^2}{2!}\Phi(\sqrt{2}a)+\frac{(vt)^3}{3!}\Phi(\sqrt{3}a)+\frac{(vt)^4}{4!}\Phi ( \sqrt{4}a)+\ldots where \Phi is the standard normal CDF. v,t,a are constants. A relevant formula is...
  16. N

    Finding CDF for Continuous RVs problem

    Homework Statement I'm given the pdf and asked to find F(y)/ cdf. I've calculated it many times, but I'm not getting the right numbers. the pdf is f(y)= .5, ....-2≤y≤0 .75-.25y,...1≤y≤3 0,...elsewhere so that means f(y)= 0,...y< -2 0.5, ...-2≤y≤0...
  17. S

    Probability problem: upper bounds on binomial CDF

    Homework Statement Hi all, just a quick question here - the setup is as follows: X is a random variable, X \sim \operatorname{Bin}(m,p) where p=2^{-\sqrt{\log n}}(\log n)^2 and m \geq 2^{\sqrt{\log n}}c for constants c, n (n "large" here). I wish to show that \mathbb{P}(X < c) \leq e^{-(\log...
  18. S

    Roll 2 fair dice, find pdf and cdf

    Hi guys. I have a test tomorrow and I am trying to work out the problems on the review sheet, but I can't figure out this one dealing with pdf and cdf. Roll 2 fair dice and call each outcome (i,j) where i,j = 1,2,...6. Define: X= {i+j if i+j ≤ 6 ; abs.value(i-j) otherwise} A) Find...
  19. Y

    Derivative of a std Normal CDF?

    I was wondering how I can find the derivative of a normal cdf with respect to a boundary parameter? I can get an answer with Mathematica or something but I have no idea how to actually do this. I don't know how fundamental theorem of calculus can be applied. (if it can be)...
  20. R

    Derivative of (1-F(x/a))(x): Interpretation & Solution

    I am trying to take a derivative of (1-F(x/a))(x) (where F(.) is a CDF and a is a parameter), and I an not sure whether the derivative should be : -(1-F(x/a))f(1/a) or -(1-F(x/a))f(x/a). Also, I am not sure how to interpret the result that at the maximum: F(x/a)f(x/a)=1.
  21. P

    Why is the cdf G considered a strictly increasing function?

    Let's say we have a cumulative distribution function (cdf) G and random numbers v1 and v2. The definition of strict increasing function is: v1 < v2 => G(v1) < G(v2). In a statistics book, the author writes: "...but with the additional assumption that the cdf G is a strictly increasing...
  22. V

    CDF of a function of 2 random variables

    Homework Statement Two toys are started at the same time each with a different battery. The first battery has a lifetime that is exponentially distributed with mean 100 min; the second battery has a lifetime that is Rayleigh-distributed with a mean 100 minutes. a) Find the CDF to the time...
  23. L

    Cdf of continous RV transformation

    Homework Statement Let f(x)= 2x , 0<x<1 , zero elsewhere, be the pdf of X. Compute the cdf of Y=1/X Homework Equations cdf of X = p(X< x) The Attempt at a Solution P(1/X <= y) = P(X <= 1/y) int 2x from 0 to 1/y = x^2 eval from 0 to 1/y = 1/y^2 so the cdf is 1/y^2...
  24. I

    Need help solving CDF (Cumulative distribution function)

    Hello, I am new. I been looking on the net for a guide how to solve the CDF by hand, i know the answer and I am about to crack this baby but I got stuck... Im trying to calculate Cumulative distribution function by hand: \int^{1}_{-1}\frac{1}{2\pi} e^{\frac{-z^{2}}{2}} dz or wolfram alpha...
  25. D

    CDF of a normal vector is there a closed form?

    Suppose a vector X of length n, where each component of X is normally distributed with mean 0 and variance 1, and independent of the other components. I want to know the probability that at least one of X_1>2, X_2>2.5, X_3>1.9, etc. happens (inclusive), i.e. the probability that the vector's...
  26. Rasalhague

    Two-tailed inverse CDF of F distribution

    Two-tailed "inverse CDF" of F distribution I'm working through Koosis: Statistics: A Self Teaching Guide, 4th edition. In Chapter 5, Koosis describes how to use a function which goes by the name of F.INV.RT(probability,deg_freedom1,deg_freedom2) in Excel 2010 to find the critical region for a...
  27. B

    Finding a combination discrete and continuous cdf to make a new cdf

    Homework Statement Let F(x)=\begin{cases} .25e^{x} & -\infty<x<0\\ .5 & 0\leq x\leq1\\ 1-e^{-x} & 1<x<\infty\end{cases}$. Find a CDF of discrete type, F_d(x) and of continuous type, F_c(x) and a number 0<a<1 such that F(x)=aF_d(x)+(1-a)F_c(x) Homework Equations The Attempt at a...
  28. V

    Trivariate CDF, solving for variable within CDF itself?

    So I've spent the better part of the last 2 days reading your forums (awesome btw) as well as as scouring Google and other sites for the past week, trying to figure out what to do. I have this equation here: p(injury) =\Phi \frac{ln(F) - 2m - 3a + b}{0.8} probability(injury) =...
  29. K

    What is the Inverse CDF of Wrapped Cauchy Distribution?

    I'm trying to calculate the inverse CDF of wrapped Cauchy distribution using Mathematica but it gets me nowhere. Probably i lack all the needed knowledge to do so (still a freshman with no statistics experience so far). Any help would be appreciated. Thanks,
  30. Andy Resnick

    CDF Collab. at Fermilab: WW/WZ Prod. Lepton+Jet Decays

    A student forwarded this to me: http://arxiv.org/abs/1104.0699 This isn't my field- can someone help walk me through it? For example, to start: "The CDF collaboration at the Fermilab Tevatron collider recently measured the cross section for the production of either an additional W or a...
  31. M

    Integral involving normal pdf and cdf

    Suppose φ(x) and Φ(x) denote the familiar standard normal pdf and cdf. I am interested in an expression for the integral: S φ(ax+b) Φ(x) dx, from ZERO to INFINITY. Many thanks
  32. I

    How to calculate confidence interval for a CDF curve

    I got a question which has been confused me for a long time. The question is to calculate the 95% confidence interval for a curve. I have already learned how to calculate for a straight line. For example, the cumulative distribution function (CDF) could be expressed as below: Y = 1/2 * {1...
  33. C

    True or False: Joint CDF Has Only One Global Max?

    True or false: "Every joint CDF has only one global maximum at F(x1*, ..., xn*) = 1? I know that the multivariate CDF is monotonically non-decreasing in each of its variables. But does that imply that it has only one global maximum? Is it possible to have two or more separate peaks where the...
  34. S

    Exploring CDF: Understanding Sample Distribution and Its Formula

    X1,X2,...,Xn is a sample from a distribution with cdf F. Show FX(i)(x)= \sum\stackrel{n}{j=i} (\stackrel{n}{j}) Fj(x)(1-F(x))(n-j) How would I start on this?
  35. M

    Prob. distribution with pmf and need cdf?

    Homework Statement X=demand for the magazine with pmf x | 1 2 3 4 p(x)| .1 .2 .4 .3 Shop owner pays $1.00 for each copy of mag. and charges $2.00. If mags. left at end of week are not worth anything, is it better to order two, three, or four copies of the mag.? I know i need to...
  36. F

    CDF of the ratio of Poisson and possibly-Poisson R.V.

    Hello, This is my first post - so let me know if I communicate incorrectly. To start, note that my thread title may be misleading as to my actual problem. I think it describes my situation, but let me provide background and then restate my problem as I see it, so as to allow for a...
  37. P

    What exactly is the difference bet. a CDF and a PDF?

    Homework Statement I understand that PDF is a derivative of CDF and hence CDF is the integral of PDF. But I don't understand the difference between PDF and simply probability? What exactly is the differece? What extra things does PDF tell us which simple probability does not? Homework...
  38. S

    Probability Question (Random Variables and CDF)

    Homework Statement A transmission channel is noisy and a binary bit (assume it is a 0 or a 1) has probability of .11 of being incorrectly transmitted. Suppose the bit is sent n (odd) times and a majority decoder announces which bit is received the majority of the time. Assume retransmissions...
  39. C

    Finding cdf and pdf of variable.

    Homework Statement Let X1 X2 X3 and X4 be four independent random variables, each with pdf f(x) = 3(1-x)2, 0<x<1, zero elsewhere. If Y is the minimum of these four variables, find the cdf and pdf of Y. The Attempt at a Solution P(Y<or= y) = 1 - P(Y>y) = 1 - P(X1>y, X2>y, X3>y...
  40. S

    How to Convert Between CDF and PDF Functions in Mathematical Equations?

    cdf to pdf and vise versa hi i'm looking for help when going from a cdf function: F(x) = { 1- e^-αx^β x ≥ 0, α>0, β>0 { 0 x < 0 to getting the corresponding pdf also i am looking to do the opposite(pdf to cdf) for: f(x) = { (1 + α)/2 for -1 ≤ x ≤ 1, -1 ≤ α ≤ 1 { 0 otherwise i'm...
  41. S

    How do I convert between cdf and pdf?

    hi i'm looking for help when going from a cdf function: F(x) = { 1- e^-αx^β x ≥ 0, α>0, β>0 { 0 x < 0 to getting the corresponding pdf also i am looking to do the opposite(pdf to cdf) for: f(x) = { (1 + α)/2 for -1 ≤ x ≤ 1, -1 ≤ α ≤ 1...
  42. E

    Exploring the CDF and MGF Relationship for Random Variables

    Hello, Suppose that the Cumulative Distribution Function (CDF) of a random variable X is F_X(x), which is by definition is: F_X(x)=\text{Pr}\left[X\leq x\right]=\text{Pr}\left[\frac{1}{X}\geq \frac{1}{x}\right]=1-\text{Pr}\left[\frac{1}{X}\leq \frac{1}{x}\right]=1-F_{1/X}\left(1/x\right)...
  43. R

    Understanding the CDF & PDF of X's Difference

    Hi guys! Homework Statement Do you know how to get the cdf and pdf of the absolute difference of two iid standard uniform X1 and X2 : X = |X1 − X2| ? Homework Equations Come across...
  44. chroot

    Obtaining samples of an RV with a given CDF

    Hey guys, I know the cdf of a random process, and I want to obtain samples of it computationally. (In MATLAB, specifically.) How can I go about doing this? The naive approach I took was to put the cdf into a variable, and then use interpolation of the inverse function. More specifically, say...
  45. M

    What is the covariance between two binomial distributions?

    Hi everybody! I have a random iid sample Xi, i=1, ..., n The empirical cdf of the sample at poin s is \hat{F}\left(s\right)=n^{-1}\sum_{i=1}^{n}{\textbf{1}_{\left(-\infty, s\right)}\left(x_{i}\right)} Clearly \hat{F}\left(s\right) is binomially distributed with parameters n and...
  46. R

    I have a very very basic question on the CDF and limits

    Homework Statement 1. For a random variable X, the function F defined by F(x) = P(X <= x),−inf < x < inf is called the cumulative distribution function of X. A property of every distribution function F is that it is right continuous with left limits. For the following functions...
  47. R

    I have a very very basic question on CDF and limits

    I am taking an "introduction to probability" class and there is a question (related to calculus) that the professor is not going to go over it. Here is the question: Well, if we look at 'a', I was trying to find the limit when x->0+ and limit when x->0-. I am getting -1 and 1 but the...
  48. E

    Order Statistics PDF and CDF

    Hello, Is the relation between the PDF and CDF of order statistics is as any PDF and CDF. i.e.: F_X(x)=\int_{-\infty}^{x}f_X(t)\,dt Regards
  49. E

    PDF and CDF Integration Simplification

    Hello, I have this equation: \int_{-\infty}^{\gamma}f_X(x)\,dx+\int_{\gamma}^{\infty}F_Y(x-\gamma)\,f_X(x)\,dx where f_X(x) and F_Y(y) are the PDF and CDF of the randome variables X and Y, respectively. Now the question is: can I write the above equation in the form...
  50. C

    Difference between a PDF and a CDF?

    I'm having a lot of trouble figuring out the difference between a probability density function and a cumulative distribution function. Is the CDF just the integral of the PDF? Thanks in advance.
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