Let Y < Y2 < Y3 < Y, be the order statistics of a random sample of size n = 4 from a distribution with pdf f(r; 0) = . 0 0. The hypothesis Ho : 0 =1 is rejected and H1 : 0 > 1 is accepted if the observed YA 2 e. Find the constant e so that the significance level is a = 0.05. & Determine the power function of the test.

Suppose that a positive continuous random variable X has probability density function given by f(x)=c(14-5x^2), where c is a constant. What is the variance of this random variable? TYPE YOUR ANSWER AS THE NEAREST NUMBER WITH TWO DECIMAL DIGITS.

Let X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use thecumulative (CDF) technique to determine the pdf of each of the following random variables: 1) Y=X^4, 2) W=e^(-x) 3) Z=1-e^(-x) 4) U=X(1-X)

Let X be a random variable with p.d.f., fx) = {2e2d-»),x > 1 0 ,x < 1. (1) Find the moment generating function of X (2) Find the mean and variance of X using the mgf from (1).

Let X ~ Uniform(a, b). Find Expected value EX.

The density of a random variable X is f(x) = C/x^2 when x ≥ 10 and 0 otherwise. Find P(X > 20).

Suppose that the lifetime, X, and brightness, Y, of a light bulb are modeled as continuous random variables. Let their joint pdf be given by:f(x,y)=λ_1λ_2e^{-λ_1x-λ_2y},x,y>0 •Are lifetime and brightness independent?•Are lifetime and brightness uncorrelated?

let Y be a single observation from a normal distribution with mean θ and variance θ^2, θ>0.

Show that the variance of a geometric random variable X; is given by V ar(X) = q/p^2 .