2. Let X ~ Exp(B), i.e. it is an exponential random variable with parameter 8. Find...
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)
4. Let X be a Exponential random variable, X ~ Expo(2). Find the pdf of X3. [Hint: pdf of XP is not (pdf of X)3, find it by differentiating the cdf of X3, i.e., Px() = P(X® S2)]
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
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.
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
Suppose that T is an Exponential (λ=1) random variable and f(x)≐⌊x⌋ i.e. is the largest integer not more than x. Find the cdf and pmf for X≐f(T). What is E[f(T)]?
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 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...