Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is expo...
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 .
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 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) = {&, «...
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 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
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 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
Let X and Y be independent exponential random variables with parameter 1. Find the joint PDF of U and V. U = X + Y and V = X/(X + Y)
Let X be an exponential random variable with parameter λ, so fX(x) = λe −λxu(x). Find the probability mass function of the the random variable Y = 1, if X < 1/λ Y = 0, if X >= 1/λ
4. (14 pts) The joint pdf of X and Y is given by: (x + cya, 0 SX S1,0 Sys1 fxy(x, y) = otherwise For this question, it may be useful draw the region in the X, Y plane where the pdf is non- zero to help you determine the limits of the integrals. (a) Find the value of the constant c. (b) Find the marginal pdfs of X and Y, respectively. (c) Find the probability that both X and...