Question

The p.m.f. of a r.v. X is 

P(x) = `{{:((2x)/(n(n + 1))"," x = 1"," 2"," ...."," "n"),(0"," "otherwise"):}` Then, E(X) = ______.

Options

• `(n+1)/3`

• `(2n+1)/3`

• `(n+2)/3`

• `(2n-1)/3`

Answer 1

The p.m.f. of a r.v. X is 

P(x) = `{{:((2x)/(n(n + 1))"," x = 1"," 2"," ...."," "n"),(0"," "otherwise"):}` Then, E(X) = `underline((2n+1)/3)`.

Explanation:

X	1	2	3	....	n
P(X)	`2/(n(n+1))`	`4/(n(n+1))`	`6/(n(n+1))`	....	`(2n)/(n(n+1))`

`E(x)=sumx_i*p(x_i)`

`=1*2/(n(n+1))+2*4/(n(n+1))+3*6/(n(n+1))+...+n*(2n)/(n(n+1))`

`=2/(n(n+1))(1+4+9+...+n^2)`

`=2/(n(n+1))(1^2+2^2+3^2+...+n^2)`

`=2/(n(n+1))*(n(n+1)(2n+1))/6`

`=(2n+1)/3`