A discrete random variable X has the PMF pXk 15 for k 0 1

A discrete random variable X has the PMF pX[k] = 1/5 for k = 0, 1, 2, 3, 4. If Y = aX + b,

(a) Find E[X].
(b) Find var(X).
(c) Find the PMF of Y ?
(d) Find E[Y ].
(e) Find var(Y ).

Solution

a) Expected value of X = ( 1/5) * [ 0 + 1 + 2 + 3 + 4 + 5 ]

= 2

b) var (X) = (1/5) * [ (0 - 2)² + (1 - 2)² + (2 - 2)² + (3 - 2)² + (4 - 2)² ]

= (1/5) * [ 4 + 1 + 0 + 1 +4]

= 2

c)

PMF ( Y = aX + b) = 1/5

d)

E (Y) = E ( aX + b) = a E(X) + b

= a(2) + b

= 2a + b

e)

Var ( Y ) = Var ( aX + b)

= a² Var(X) + var(b)

= a² (2)

= 2a²