Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with...
Problem 8: 10 points Suppose that (X, Y) are two independent identically distributed random variables with the density function defined as f (x) λ exp (-Ar) , for x > 0. For the ratio, z-y, find the cumulative distribution function and density function.
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common density function f(x) 2x 0〈x〈1 0 otherwise Find the probability density function of X Y. 42.5 If 0 < a < l then ÍxHY(a) 2a3. If 1 < a < 2 then ÍxHY(a) -릎a3 + 4a-3. If a 〉 2 then fx+y(a) 0 and 0 otherwise.
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
Let h be an exponentially-distributed random variable with the distribution function p- exp(-x) for x > 0 and ph = nction Ph 0 for a s 0. Derive the distribution function of its square root, Solution: 2y exp(-y2
Let Y1, Y2, . .. , Yn be independent and identically distributed random variables such that for 0 < p < 1, P(Yi = 1) = p and P(H = 0) = q = 1-p. (Such random variables are called Bernoulli random variables.) a Find the moment-generating function for the Bernoulli random variable Y b Find the moment-generating function for W = Yit Ye+ … + . c What is the distribution of W? 1.
Let U., Un be independent, identically distributed Uniform random variables with (continu- ous) support on (0, b), where b >0 is a parameter. Define the random variable Y :--Σίι log(U), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b of Y by explicitly computing it.
1. [26 pts Let Uı, , Un be independent, identically distributed Unifomn random variables with (continu- ous) support on (0, b), where b> 0 is a parameter. (a) Define the random variable Y :--Σί 1 log(U,), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b) ofY by explicitly computing it (b) Based on the pdf you found in part (a) above, determine the third moment of Y, i.e., EY] (c) Suppose now that...
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
Problem 7: 10 points Consider independent random variables, [Y: 1, 2, ..., ), having the same Gamma distribution, with the density, .узе-2y for y > 0 Suppose that a random variable, N, does not depend on all Y, and is geometrically distributed, so that PIN = n] = n, for n=1, 2, Consider a random sum, S Y 1. Determine the marginal expectation of S. 2. Determine the marginal variance of S.