Given the following probability mass function (PMF) of a random variable X, i.e.

Question

Given the following probability mass function (PMF) of a random variable X, i.e., Find the parameter p 's valid range such that the graph above is a true PMF. What kind of distribution dose the random variable X fall into? Find the following statistics: The expectation E(X) = ? The second moment E(x^2) = ? The variance Var(x) ? If we linearly transform the r v.X, resulting in a new r. v. Y, say Y = 2x - 1. Find the distribution (PMF) of Y. Find the values of E(Y), E(Y^2), and Var(Y). What if the transform is more general, say Z = ax + b, where the parameters a notequalto 0, b are deterministic constants. Find the distribution (PMF) of the r. v. Z and the values of E(z) E(Z^2), and Var (Z).

Explanation / Answer

(1) 0 < p < 1

(2) Bernaulli distribution

(3) E(X) = 0.p + 1.(1-p) = 1-p

     E(X2) = 02.p + 12.(1-p) = 1-p

     Var(X) = E(X2) - (E(X))2 = 1-p - (1-p)2 = (1-p)(1-(1-p)) = (1-p).p

(4) Consider Y=2X-1

     When X=0, Y=(2.0)-1 = -1 with probability p

     When X=1, Y=2.1-1=2-1=1 with probability 1-p

So pmf of Y is

   Y = -1, with prob. p and 1 with prob. 1-p

E(Y) = -1.p + 1.(1-p) = 1-2p

E(Y2) = -12.p + 12.(1-p) = p+1-p=1

Var(Y) = 1- (1-2p)2 = 4p - 4p2 = 4p(1-p)

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