Let px c23x x123 Infinitely many values of x Find the const

Let p(x)= c(2/3)^x , x=1,2,3 (Infinitely many values of x)

Find the constant c so that p(x) becomes a pdf of some r.v. X.

Solution

Note that the sum of the probabilities from x = 1 to infinity is 1.

Hence,

c(2/3)^1 + c(2/3)^2 + c(2/3)^3 + c(2/3)^4 + ...

which is a geometric series of first term A1 = c(2/3) and common ratio r = 2/3. Hence,

Sum = A1/(1-r)

=c(2/3)/(1-2/3)

=c(2/3)/(1/3)

=c(2) = 1

Hence,

c = 1/2 [ANSWER]