The probability function fX for a random variable X is given by the table below.

Question

The probability function fX for a random variable X is given by the table below.

x | -1 0 1 2 3

fX(x) = P(X = x)| 0.2 0 u 0.2 0.1

(a) Find u.

(b) Find the expected value and variance of the distribution for the random variable X: E(X) and Var(X).

(c) Assume now Y = X − 2, then you may regard Y as another random variable. Find the expected value and variance for the distribution of this random variable Y : E(Y ) and Var(Y ).

(d) No need to answer this part. Just a remark for your interest: In part (c), you are finding E(X − 2) and Var(X − 2). It would be interesting to compare these values with E(X) and Var(X), and observe that E(Y ) = E(X) − 2 and Var(Y ) = Var(X).

Answer 1

(a) Since f(x) is a probability distribution, the probabilities given to each value of the random variable in the table must sum to 1. Therefore, we have 0.2+0+u+0.2+0.1 = 1. Solving this equation yields u = 0.5.

(b) E(X) = (-1)(0.2)+(0)(0)+(1)(0.5)+(2)(0.2)+(3)(0.1) = 1

Var(X) = E(X²) - E(X)² = (-1²)(0.2)+(0²)(0)+(1²)(0.5)+(2²)(0.2)+(3²)(0.1) - 1² = 1.4

(c) E(Y) = E(X-2) = E(X) - 2 = 1 - 2 = -1

Var(Y) = Var(X-2) = Var(X) = 1.4

Hope this helps!