Exercise 3. (QUANTIZATION, FROM TEXTBoOK, PROBLEM 4.61) Let X be an exponential random variable with parameter...
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...
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
Show all details: Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as
3. If X is an exponential random variable with parameter λ > 0, show that for c > 0 cX is exponential with parameter λ/c.
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
Exercise 7. Let X and Y be A. independent exponential random variables with a common parameter (1) Find the transform associated with aX +Y, where a is a constant. (2) Use the result of part (1) to find the PDF of aX +Y, for the case where a is positive and different than1 (3) Use the result of part (1) to find the PDF of X-Y. Justify your answers. Exercise 7. Let X and Y be A. independent exponential random...
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)
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)