Statistics and Probability Answers

You are asked to find the mean and variance of a random variable Y whose distribution has probability generating function is

qr /1 − (1 − q)r. 0 < q < 1 

 Suppose that X and Y have bivariate normal distribution with the probability density function f(x1, x2) = k exp − (8X^2 − 6XY − 18Y^2 ) Find (a) Pr(X + Y > 1/2) (b) the joint moment generating function of Z1 = 2X − XY and Z2 = 3X + 2Y

Let X,Y be a random sample from the distribution

f(x) = 1 0 < x < 1

Further let Y = max(X,Y). Using the distribution function technique,

find the probability density function of Y. Hence find its mean and variance values.

Let X1, X2 have joint probability density function

f(x1, x2) = {

1/8e

−(8x1+x2)

, x1,x2>0

0, elsewhere

Find the probability density function of Y =1/2 (X1 + X2).

1.     The time to complete the production of certain product for two machines produced by two different

well−known companies (company A and B) were observed to be completely different. The time machine from company A takes to complete the production (in hours) is X~Exp(2) and the time machine from company B takes to produce the product (in hours) is Y~Unif(0, 1). If the performances of the two machines is assumed to independent, what is the distribution of Z = X + Y, the total time they take to complete the production of the product? Hint: use the convolution method.