If the range ot a random vaniabie `X` is `0,1,2,3,` at `P(X=K)=((K+1)/3^k)` a for `k>=0,` then a equals

Options

A. `(2)/(3)`
B. `(4)/(9)`
C. `(8)/(27)`
D. `(16)/(81)`

Answer

B

Explanation

Given, `P(X=k)=((k+1)a)/(3^(k))`, for
`x in {0,1,2, . . .,oo}`
As we know that,
`P(0)+P(1)+P(2)+. . . Oo=1`
`impliesa+(2a)/(3)+(3a)/(3^(2))+ . . . Oo=1`
Let `S=a(1+(2)/(3)+(3)/(3^(2))+(4)/(3^(3))+ . . .oo)`
`((1)/(3)S=a((1)/(3)+(2)/(3^(2))+(3)/(3^(3))+ . . .oo))/(S-(1)/(3)S=a(1+(1)/(3)+(1)/(3^(2))+(1)/(3^(3))+ . . . oo))`
`implies(2)/(3)S=a((1)/(1-(1)/(3)))`
`impliesS=(9a)/(4)`
`therefore` From eq. (i), we get
`(9a)/(4)=1impliesa=(4)/(9)`