(X, Y, Z, W) have multinomial distribution: f(x, y, z, w) = n! x!y!z!w! 01020 (1-01-02-03)", x = 0, 1, 2, ..., n, y = 0, 1, 2,...,n, z = 0, 1, 2,...,n, w = 0, 1, 2,...,n, x+y+z+w = n. Then E(YX = x) is (A) a + β X (B) a + ẞx (C) a + bx+yz (D) a + 3x + y z + C w

If X is a continuous random variable with the uniform distribution on [0,1], then what is P[X = .5]? Explain.

Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.

If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.

Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).

Let (x1, x2, ..., xn) be independent samples from the uniform distribution on [0, θ]. LetX(n) and X(1) be the maximum and minimum order statistics respectively,(a) show that 2nY~X22 where Y=-In(x(n)/θ)

2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter l.   a)     If X1, X2, . . . , Xn are the times, in minutes, between successive customers selected randomly, estimate the parameter of the distribution. b)     b) The randomly selected 12 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c)      In order to estimate the distribution parameter with 0.3 error and 4% risk, find the minimum sample size.

If X has an F distribution with ν1 and ν2 degrees offreedom, show that Y = 1Xhas an F distribution with ν2 and ν1 degrees of freedom.

Suppose X has uniform distribution on (−1, 1) and, given X = x, Y is uniformly distributed on  (−(1-x^2)^1/2, (1+x^2)^1/2). Find the join denisty of X and Y.

IF F(x, y) is the value of the joint distribution function of X and Y at (x, y), show that the marginal distribution function of X is given by   G(x) = F(x, ∞) for - ∞ <x < ∞   Use this result to find the marginal distribution function of X for the random variable   F(x, y) = { (1-e-x^2 ) (1- e-y^2) for x>0, y> 0 and 0 elsewhere