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- If X is a Poisson variable such that P(X =2) = 9P(X= 4) + 90P(X = 6), find the mean and variance of X.If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.A random variable X has only two values a and b with P(X = a) = p , P(X = b) = q (p + q = 1).Find its mean value and variation.
- A random variable X takes values 0,1,2 where E(x) =1.5, var(x)=0.25 then E (3x– x² +4) =X is a discrete uniform (1,3) random variable. When X=x, Y is discrete uniform (1,x). (a) Find the correlation of X and Y (b) Find the mean value of X when Y=2.18 Given o = (aX + 3Y) where X and Y are zero - mean random variables with variances o =4 and o= 16. Their correlation coefficient is p=- 0.5 %3| (a) Find a value for the parameter 'a' that minimizes the mean value of w (b) Find the minimum mean value.
- B) Let the random variable X have the moment generating function e3t M(t) for -1Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the aboveConsider the random variable X such that E(X) = 3 and Var(X) = 2 if the random variable Z = 1+5X. E(Z) is what?2. A random variable X follows Beta(2,3) distribution. (a) Show that the cumulative distribution function F of a Beta(2,3) random variable is F(x) = 6x? 823 + 3x4, r € [0, 1]. (b) Find the sampling distribution (pdf) of the median for a random sample of size n = 2m +1 from Beta(2,3).Example 24: If a random variable X has the moment generating function Мx (t) — 2-t' determine the variance of X.Let X be a discrete random variable with P(X = -1) = .3, P(X = 1) = .45, P(X = 2) = .15 and P(X = 6) = .1. Find Var(X).