5 20 points The probability generating function pgf of a ran

5. (20 points) The probability generating function (p.g.f) of a random variable X is known to be t(1- t6) P(t) 6(1- t) Find the probability function (p.d.f of X

Solution

The probability generating function can be obtained from the probability density function as follows:

P(t)=(n=0->) f(n) t^n -------------------------------------------->1

where f is PDF.

P(t) = t (1-t^6)/6 (1-t)

= sigma(n =1 to 6) 1/6 * t^n

= sigma (n =1 to 6) f(n) * t^n ------------------------------------->2

Comparing two equations 1 and 2

PDF f(n) = 1/6