Let R be a random variable with an exponential distribution with parameter α = 1. Conditional on R, Y has pdf
fy|R (y|r) = { ry 0<=y<= sqrt(r/2); otherwise 0
Determine the conditional pdf of R given Y .
Let R be a random variable with an exponential distribution with parameter α = 1. Conditional...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of a toss of a fair coin Compute the CDF and the PDF of Z = XY Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of...
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
Let X be an exponential random variable with parameter 1 = 2, and let Y be the random variable defined by Y = 8ex. Compute the distribution function, probability density function, expectation, and variance of Y
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF Problem 5. Let X...
Let X and Y be a random variable with joint PDF: fxx (x, y) = { 1, 2 > 1,0 Sysi 0 otherwise 1. What is a? 2. What is the conditional PDF fy|x(x|y) of Y given X = x? 3. What is the conditional expectation of Ygiven X? 4. What is the expected value of Y?
The PDF of random variable X and the conditionalPDF of random variable Y given X are fX(x) = 3x2 0≤ x ≤1, 0 otherwise, fY|X(y|x) = 2y/x2 0≤ y ≤ x,0 < x ≤ 1, 0 otherwise. (1) What is the probability model for X and Y? Find fX,Y (x, y). (2) If X = 1/2, nd the conditional PDF fY|X(y|1/2). (3) If Y = 1/2, what is the conditional PDF fX|Y (x|1/2)? (4) If Y = 1/2, what is...
Let X and Y be a random variable with joint PDF: f X Y ( x , y ) = { a y x 2 , x ≥ 1 , 0 ≤ y ≤ 1 0 otherwise What is a? What is the conditional PDF of given ? What is the conditional expectation of given ? What is the expected value of ? Let X and Y be a random variable with joint PDF: fxv (, y) = {&, «...
I. Let X be a random sample from an exponential distribution with unknown rate parameter θ and p.d.f (a) Find the probability of X> 2. (b) Find the moment generating function of X, its mean and variance. (c) Show that if X1 and X2 are two independent random variables with exponential distribution with rate parameter θ, then Y = X1 + 2 is a random variable with a gamma distribution and determine its parameters (you can use the moment generating...