What is Density function: Definition and 192 Discussions
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.
The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.
Homework Statement
Let X and Y be independent random variables each having the Cauchy density function f(x)=1/(∏(1+x2)), and let Z = X+Y. Show that Z also has a Cauchy density function.
Homework Equations
Density function for X and Y is f(x)=1/(∏(1+x2)) .
Convolution integral =...
Homework Statement
The probability density function of a random variable y is:
f(y) = 100ye^{-10y}, if y>0
f(y) = 0 otherwise
What is the probability that 45y <= 10?
Homework Equations
E(y) = ∫yf(y)dy
Var(y) = ∫(y-E(y))f(y)dy
The Attempt at a Solution
I...
Homework Statement
If U is uniform on [−1, 1], find the density function of U^2.
Homework Equations
f(u) = 1/(b-a)
The Attempt at a Solution
I actually solved the problem already, but I am having trouble defining what the boundaries are for U^2. My work is uploaded in paint...
Homework Statement
Let f(x,y)=xe^{-xy} x \geq 0, y \geq 1
is this a probability density function? If not, find a constant that makes it a pdf.
Homework Equations
To be a pdf, we must have \int_1^\infty \int_0^\infty \! xe^{-xy} \, \mathrm{d} x \mathrm{d} y=1
The Attempt at a...
Homework Statement
Random voltage is defined with its probability density function:
p_{\xi}(v)=2,25u(v+2)e^{-3(v+2)}+k\delta (v-2)
u-Heaviside step function
a) Find constant k.
b) What is the probability of a random variable to take value of 2.
c) Find the cumulative distribution function...
For the density function for random variable Y:
f(y) = cy^2 for 0<= y <= 2; 0 elsewhere
We are asked to find the value of c. I did a definite integral from 0 to 2 of cy^2. I get c = 3/8. Why would the book show an answer of c = 1/8? Is this an error on their part or am I missing something...
Hello everyone!
I am stuck in my research with a probability density function problem..
I have 'Alpha' which is a random variable from 0-180. Alpha has a uniform pdf equal to 1/180.
Now, 'Phi' is a function of 'Alpha' and the relation is given by,
Phi = (-0.000001274370471*Alpha^4) +...
My goal here is to at least approximately calculate the probability density function (PDF) given the moment generating function (MGF), M_X(t).
I have managed to calculate the exact form of the MGF as an infinite series in t. In principle, if I replace t with it and perform an inverse...
Hello! I have been having problems with understanding how the probability density function is calculated. However, at the same time, I need it urgently for my research. Well, you could start by giving me a definition of
1. Refernce measure
2. That 'E' sign(looks like an epsilon, and I sound...
hey guys, i am really confused on something.here is the thing:
i have;
i=x+(x^2-y)^(1/2)
and here x is uniform distribution on (a,b)
y is uniform distribution on (c,d)
x and y independent
i need to find the probability density function of i but how?
actually i don't know how to...
Hi guys i am struggling in how to find the density function for X^2Y^2 and max(X,Y).
Here's the scenario:
Suppose a random variable X has the Uniform distribution on the interval [-1,1].
Suppose a random variable Y has the exponential distribution with parameter lamda=2.
X and Y are...
Homework Statement
Is the PDF of something between two different bases or wavefunctions always 0?
For instance, if you have the lowering operator \hat{}a -
<n|\hat{}a|n>
that changes to <n|\sqrt{}n|n-1> =0
I'm not sure I understand the physical scenario if this is true however.
The energy density function for the electric field in vacuum is
u=\dfrac{\epsilon_0}{2}E^2
And the cited textbook result for the energy density inside a dielectric is:
u=\dfrac{\epsilon_0 \epsilon_r}{2}E^2
Now, one way to reach the upper formula is to look at the energy as...
Hello,
given a continuous random variable x with a known PDF, how can we determine in general the PDF of the transformed variable f(x) ?
For example f(x)=x+1, of f(x)=x2 ... ?
Also, if we have two random variables x,y and their PDF's, is it always impossible to determine the PDF of f(x,y)...
Problem :
A random variable X has probability density function
f(n) =
\begin{cases}
Ax, & \mbox{0 lessthan equal x less than 5 } \\
A(10-x), & \mbox{5 less than equal x less than 10 }\\
0, & \mbox{otherwise }
\end{cases}
i)Determine A
ii)Find P(2.5 <= X <= 7.5)
Solutions
i) let P(A)...
Homework Statement
X and Y are random variables with the joint density:
fXY(x,y) = k*e^(-lambda * x) if 0 < y < x < infinity
= 0, otherwise
Find P(X + 2*Y <= 3)
Homework Equations
I found k = lambda^2
The Attempt at a Solution
I'm not sure exactly how to solve this, but...
Continuous random variable X is defined in the interval 0 to 4, with
P(X>x)= 1- ax , 0<=x<=3
= b - 1/2 x , 3<x<=4
with a and b as constants. Find a and b.
So the area under the pdf is 1, then i integrated both functions and set up my first equation.
Next, it seems that the...
Homework Statement
Two random variables, x and y. Density functions are m(x) and f(y), respectively. x is defined on [0,1] and y on [0,1-\rho]. I also know that
\int_{0}^{1} m(x) \,dx = 0.5
\int_{0}^{1-\rho} f(y) \,dy = 0.5
Knowing that f(y) is essentially a transformation of m(x)...
I have a region R = {(x,y): |x| + |y| <= 1}
I have sketched the graph on the x-y plane and shaded in the region R (I got a diamond).
How do I find the area of R and hence find the joint density function f(x,y) and how do I specify the region for which f(x,y) is non-zero.
Homework Statement
Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X.
Homework Equations
f(x,y) = 2 for 0 < x < y < 1
f(x,y) = 0 otherwise.
The Attempt at a Solution
For Z = X + Y, I can derive the fact that,
f_Z(z) =...
Homework Statement
A point Q is chosen at random inside the unit square. What is the density function of the sum of the coordinates of point Q? What is the density function of the product of the coordinates of the point Q? Use geometry to find these densities.
Homework Equations
P(a <...
Homework Statement
Let X be a posative random variable with probability density function f(x). Define the random variable Y by Y = X^2. What is the probability density function of Y? Also, find the density function of the random variable W = V^2 if V is a number chosen at random from the...
Homework Statement
Roll a fair die three times
Let X be the number of different faces shown all together ( X = 1,2,3 )
Find px(k)
Homework Equations
The Attempt at a Solution
Alright so I kno that i need to get the individual probabilities of each outcome
The first one where...
p(t) = Ce-Ct
i know P(t) is the fraction of the group surviving t years or less
now the question is: suppose a patient has a 70% probability of surviving at least 2 years, find C.
to find c i would take the integral of p(t) but what would my limits be? surviving 2 at least 2 years, so...
Homework Statement
Let X and Y have JD f(x,y) = e^-y, 0<x<y
Find:
a) E(X|Y=y), E(Y|X=x)
b) density function of R.V. E(X|Y), E(Y|X)
The Attempt at a Solution
a)
I have found E(X|Y=y) = y/2 for y>= 0
E(Y|X=x) = x +1 for x>= 0
by finding fx(x) = ∫(x to infinity) e^-y dy = e^-x...
Y-U(-2\pi,2\pi)
find the density function of z=tan(Y)
?
i had a similar question
X-U(0,1)
find the density function of W=a+bx
the solution is
W-U(a,a+b)
how to solve the first question ??
Homework Statement
Let f(x) = (1 + ux)/2 for -1<= x <= 1 and 0 otherwise . where -1<= u <= 1
a) show f is a density
Homework Equations
TO show
1. f(x) >= 0
2. intergeral f (from -infinity to infinity) = 1
The Attempt at a Solution
I have done 2. and proved that it is 1...
Hi everyone,
I have a simple question (assuming since it was only worth 5% of total marks in the exam) about the PDF of a random variable.
Given that PDF of random signal equals p(X) = \Lambda(X), where \Lambda is the triangle function, what would be the PDF of the random signal Y, Y = -3X...
Could anyone help me figure out the the probability density function (pdf) of |X|^(1/2)+|Y|^(1/2)+|Z|^(1/2) if X, Y and Z are distributed normally with mean 0 and variance 1, N(0,1) ?
Thanks in advance.
Homework Statement
Let X and Y be independent uniform (0,1) random variables.
a. find th ejoint density of U=X, V=X+Y.
b. compute the density funciton of V.
Homework Equations
The Attempt at a Solution
Part a. is not a problem. I don't understand how the bounds for part b...
Density Function and E(x)[solved]
The density function of X is given by
https://webwork.math.lsu.edu/webwork2_files/tmp/equations/48/83b2bf602cc895a007a673a9a23c3c1.png
If the expectation of X is E(X)=-1, find a and b.
The Attempt at a Solution
I'm actually working ahead of the...
Homework Statement
Probability of a car starting up is 0.9
Probability of a car NOT starting up is 0.1
Cars are tested until 2 functional cars are found.
Find Bernoulli probability function associated and PROVE that it is a pdf (probability density function).
Homework Equations...
I'm just curious as to how to prove that a Bernoulli distribution probability function is valid (ie. that it is indeed a probability distribution function). I have a hunch that all we do is add up all of the probabilities associated to every x value, but I'm not sure. Can someone confirm this...
Homework Statement
There are two parts to question, the first asks for you to find the moment of inertia I for a thin disk of uniform density, a relatively trivial problem.
My problem centers around that second part, "Repeat the case where the density increases linearly with r, starting at...
Homework Statement
f(x) is a density on R+ so f(x) < 0 if x < 0. Define g_(X,Y)(x,y) = f(x+y)/(x+y). Show g is a density on R^2.
Homework Equations
the first part is easy (showing that g is in fact >= 0. The part I am struggling with is the double integral of the g(x,y) over the positive...
Homework Statement
Consider a standard i.i.d. Gaussian random vector \mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T and its squared magnitude
||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2.
According to my textbook, to derive the density function of a chi-square random variable ||\mathbf{X}||^2 , one...
According to my textbook, to derive the chi-square density function, one should perform three steps. First we consider a standard i.i.d. Gaussian random vector \mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T and its squared magnitude
||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2.
1. For n = 1, show that...
Probability density function after filtering
Hello,
I am trying to find how a random variable will transform once gone through
a filter.
For example, I have a random sequence x(t), going through a filter h(t). Thus,
y(t) = x(t)*h(t) ; % '*' is convolution.
Now I want to find out how...
I am tyring to solve the follwing problem...
http://www.imagedump.com/index.cgi?pick=get&tp=549226
What is the appropriate K valuefor this to be a legitimate probability density function?
Im not exactly sure of the approach to this problem...
Homework Statement
Consider the wave packet defined by
psi(x) = integral(limits of +infinity and - infinity) dke^(-alpha(k-k_0)^2) e^(ikx)
a)What is the mean value of the momentum p barred (it's just a line over the p) of the particle in the quantum state given by this wave function...
A point is selected at random and uniformly from the region
R = {(x,y): |x| + |y| <= 1 }
Find the probability density function of the x-coordinate of the point selected at random.
By definition f(x) = the integral of f(x,y) over all y values.
after this I'm pretty much stuck...
Homework Statement
Suppose that a point (X_1 , X_2 , X_3) is chosen at random, that is, in accordance with the uniform probability function over the following set S:
S = {(x_1, x_2, x_3) : 0 \leq x_i \leq 1 for i =1,2,3}
Determine P[(X_1 - 1/2)^2 + (X_2 - 1/2)^2 + (X_3 - 1/2)^2) \leq...
Homework Statement
Let X be a random number from (0,1). Find the probability density function of Y = 1/X.
Homework Equations
The Attempt at a Solution
I keep thinking this is easier than I am making it out to be, but the only places I find anything similar searching is on two...
How to find the probability density function of a simple harmonic oscillator? I know that for one normal node is should be a parabola but what is the formula and how do we derive it?
Please help me with this. Any suggestions are greatly appreciated.
Imagine that I have a bank account. X is the amount of cash on the account at time t+1. Y is the amount of cash at time t. The amount of cash depends on the deposits made and on the amount of cash during the previous period...
Hello!
I'm taking a mathematics course in probability and stochastic processes and we started covering the CDF (cumulative distribution function) which i understand perfectly and then the PDF (probability density function). The PDF was defined to be the derivative of the CDF. Now, the CDF is...
Suppose that h is the probability density function of a continuous random variable.
Let the joint probability density function of X, Y, and Z be
f(x,y,z) = h(x)h(y)h(z) , x,y,zER
Prove that P(X<Y<Z)=1/6
I don't know how to do this at all. This is suppose to be review since this is a...
Assume that two random variables (X,Y) are uniformly distributed on a circle with radius a. Then the joint probability density function is
f(x,y) = \frac{1}{\pi a^2}, x^2 + y^2 <= a^2
f(x,y) = 0, otherwise
Find the expected value of X.
E(X) = \int^{\infty}_{- \infty}\int^{\infty}_{-...