Let a 2. Suppose a discrete random variable X has the following probability
distribution:
p ( 0 ) = P ( X = 0 ) = c,
p (k) = P ( X = k ) = 1/(a^k), k = 1, 2, 3, … .
a) Find the value of c ( c will depend on a ) that makes this is a valid probability
distribution.
b) Find the moment-generating function of X, M( t ). For which values of t does it exist?
c) Find E ( X ).

Question

Let  a > 2.  Suppose a discrete random variable X has the following probability
distribution:
p ( 0 ) = P ( X = 0 ) = c,
p (k) = P ( X = k ) = 1/(a^k), k = 1, 2, 3, … .
a) Find the value of  c  ( c will depend on a )  that makes this is a valid probability
distribution.
b) Find the moment-generating function of  X,  M( t ).  For which values of  t  does it exist?
c) Find  E ( X ).

tkhunny · Answer

c + 1/a + 1/(a^2) + 1/(a^3) + ...

Can you add those up?

Hint: 1/(1-x) = 1 + x + x^2 + ...