2) A uniformly distributed continuous random variable, X, is described by the PDF shown. It is input into a system producing a new random variable Y, described by the equation shown. Determine an expression for the CDF for Y as a function of fx(x). You do not need to solve non-trivial integrals. Just set them up. fx(x)* 1/2 2 x 1 y= g(x)=3x+1 x

If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.

Please write each step.

Suppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where u(x) is the unit step function. Determine the conditional pdf fx(x 1

Express var(X + Y), var(X − Y), and cov(X + Y, X −Y) in terms of the variances and covariance of Xand Y.

If E(X) = 6 and E[(X)(X-1)] = 51,  a) find the second moment  b) find the variance of X Please provide your working and round your answers to 3 decimal places

Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y

The density function of X is -120x- 0, f(x) = (2(x - 1), (a) Find the mean E(4X+3); (b) Find the variance 3x+2. 1 < x < 2, elsewhere.

A fair coin is tossed three times and the random variable x equals the total number of heads. Find the sketch F(x) and f(x).

Denote owl and wood rat populations in a particular area at time k by xk LK Rk where k is the time in months, Lk is the number of owls in the area studied, and R is the number (in thousands) of wood rats in the area studied. Suppose Lk+ 1 = (0.2)Lk + (0.6)Rk and Rk+1=(-0.1)Lk + (0.9)Rk. Determine the evolution of this system. (Give a formula for xk.) Does the owl population grow or decline? Does the wood rat population grow or decline?