= 2 3. Moment Generating Function Let X be a discrete random variable such that 4 P(X = 6) = P(X = 2) = Find the moment generating function of X. 1 P(X = 0) P(X = == -3) = 20 2

A random variable X takes values 0,1,2 where E(x) =1.5, var(x)=0.25 then E (3x– x² +4) =

Determine ?(?>2).

Example 24: If a random variable X has the moment generating function Мx (t) — 2-t' determine the variance of X.

Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.

show all solution 1. Suppose X is a random variable with E(X2)=10 and E(X)=2. Find the following: a) E(2X + 5) b) Var(X) c) Var(2X + 5) d) Var(2X – 5)

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

Let X1, X2, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?

(4) If a point x is taken at random on the interval [0, 2a] and all the points are equally likely to be taken, the probability that the product x(2a-x) > is