Question

2.1 Let X be a discrete random variable with the following probability distribution

Xi	0	2	4	6	7
P(X = xi)	0.15	0.2	0.1	0.25	0.3

a) find P(X = 2 given that X < 5)

b) if Y = (2 - X)² ,

i. Construct the probability distribution of Y.

ii. Find the expected value of Y

iii. Find the variance of Y

Answer 1

a) P(X = 2 | X < 5) = P(X = 2 and X < 5) / P(X < 5)

                             = P(X = 2) / P(X < 5)

                             = (0.2) / (0.15 + 0.2 + 0.1)

                             = 0.4444

b) Y = (2 - X)²

i)

X	Y	P(x)
0	4	0.15
2	0	0.2
4	4	0.1
6	16	0.25
7	25	0.3

Y	P(Y)
0	0.2
4	0.1+0.15 = 0.25
16	0.25
25	0.3

ii) Expected value = E(Y) = 0 * 0.2 + 4 * 0.25 + 16 * 0.25 + 25 * 0.3 = 12.5

iii) E(Y²) = 0² * 0.2 + 4² * 0.25 + 16² * 0.25 + 25² * 0.3 = 255.5

Var(Y) = E(Y²) - (E(Y))² = 255.5 - 12.5² = 99.25