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. All discrete random variables must equal 1 when adding together the probabilities associated with their respective values. Therefore, 0.2 + 0 + u + 0.2 + 0.1 = 1, and u = 0.5

b. For any discrete random variable, the equation for the mean or expected value E(X) = Σx_if(x_i) or E(X) = 0.2*-1 + 0*0 + 0.5*1 + 0.2*2 + 0.1*3 = 1

The equation for the variance V(X) = Σ(x-mean of x)²/N or V(X) = (-1-1)² + (0-1)² + (1-1)² + (2-1)² + (3-1)² = 10

c. If Y = X - 2, then the distribution of X is being translated to the left. In other words, the values of the distribution of X are now (in Y) -3, -2, -1, 0, 1. For the expected value, translations to the left means decreasing the mean of X by 2 (in this case). Thus E(Y) = -1. The variance is unaffected by translations and thus V(Y) = 10.