Question

The probability distribution of a discrete random variable X is given by: P(X = −1) = 1 5 ,P(X = 0) = 2 5 and P(X = 1) = 2 5 (a) Compute E[X]. (b) Determine the probability distribution Y = X 2 and use it to compute E[Y ]. (c) Determine E £ X 2 ¤ using the change-of-variable formula. (You should match your answer in part (b). (d) Determine V ar (X).

Answer 1

Explanation:

Given that the probability distribution of a discrete random variable X is given by:
`P(X = −1) = (1)/(5) \\ P(X = 0) =(2)/(5)\\ P(X = 1) = (2)/(5)`

a)
`E(x) =\Sigma x_i p_i = -1((1)/(5) )+0(((2)/(5) )+1(((2)/(5) )\\=((1)/(5) )`

b)
`Y=x^2`

So y can take values as 0 and 1.

P(Y=0) =
`(1)/(5)`

P(Y=1) =
`(4)/(5)`

(since -1 also becomes +1 when squared)

E(Y) =
`0((1)/(5))+1((4)/(5))\\=(4)/(5)`

c)
`E(x^2)=\Sigma x^2 p = (4)/(5)`

d) Var(x) =
`(4)/(5) -(1)/(25) =(19)/(25)`