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 "k in {0,1,2,...oo}`
Clearly,
`P(0)+P(1)+P(2)+...oo=1`
`rArra+(2a)/3+(3a)/3^2+...oo=1`
Let `S=a(1+2/3+3/3^2+4/3^3+...oo)`
`rArr" "1/3S=a(1/3+2/3^2+3/3^3+...oo)`
`therefore" "S-1/3S=a(1+1/3+1/3^2+1/3^3+...oo)`
`rArr" "2/3S=a(1/(1-1/3))rArrS=(9a)/4`
`therefore" From Eq. (i) "(9a)/4=1rArra=4/9`