PROBLEM 5: Suppose that 0 x b < a are positive constants. Furthermore, the random variable...
5. (Discrete and ontinuous random variables) (a) Consider a CDF of a random variable X, 10 x < 0; Fx(x) = { 0.5 0<x< 1; (1 x > 1. Is X a discrete random variable or continuous random variable? (b) Consider a CDF of a random variable Y, 1 < 0; Fy(y) = { ax + b 0 < x < 1; 11 x >1, for some constant a and b. If Y is a continuous random variable, then what...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
4. Suppose that X is a random variable such that P(X < 0) = 0. You toss a fair coin and if the head comes up, you define Y to be VX; if the tail comes up, you define Y to be - VX. a. Find the cumulative distribution function of Y in terms of the cumulative distribution function of X. (You will probably want to consider two cases, one for y<0 and the other for y> 0.) b. Now...
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)
Problem # 10, Let X be a random variable with CDF: 0 (x + 5)2/144-5 < x < 7 Ex (x) = X(r Find E(X], , and E[X"].
PROBLEM 4 Let X be a continuous random variable with the following PDF 6x(1 - 1) if 0 <r<1 fx(x) = o.w. Suppose that we know Y X = ~ Geometric(2). Find the posterior density of X given Y = 2, i.e., fxy (2/2).
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.
1. Suppose X ∼ Gamma(a,b) and Y ∼ Gamma(c,d). Furthermore suppose X and Y are independent. Let W = X + Y . (a) Find the MGF of W. (b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given X-k, for some k-n. Let a = P(X-k), and suppose that the value of a has been computed (a) Give the inverse transform method for generating Y. (b) Give a second method for generating Y (c) For what values of a,...
5. (6 pts.) Suppose the random variable X has probability density function ()if 0, ifx<0. Find the mean and variance of X. Write the equation one needs to solve to find the median m of X. Do not solve the equation, but at least simplify it enough so that it does not involve any integrals.