Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter...
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...
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 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...
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...
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
6. Let X be an exponential random variable with parameter 1 = 2. Compute E[ex]. = 7. Consider a random variable X with E[X] u and Var(X) 02. Let Y = X-4. Find E[Y] and Var(Y). The answer should not depend on whether X is a discrete or continuous random variable.
3. Let T be an exponential random variable with parameter 3 and let W be a random variable independent of T which assumes the value 1 with probabil ity 2/3 and the value -1 with probability 1/3. Find the density of X = WT Hint: It would help to split up the event {X < x} as the union of {X < X x, W 1} . (10 points)
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 # 8. a) Let X be a continuous random variable with known CDF FX(x). LetY = g(X) where g(·) is the so-called signum function, which extracts the sign of its argument. In other words, g(X) = { -1 x<0, 0 x=0, 1 x>0 } Express the PDF fY (y) in terms of the known CDF FX(x). b) Let X be a random variable with PDF: fX(x) = { x/2 0 <= x < 2, 0 otherwise} Let Y be...