I have gotten the first half of this question, but need help on the second half.
Let be a random variable with geometric distribution of parameter .
a) Show that satisfies the memoryless property:
Here is my answer to this half:
If has the probability distribution function ,
Consider the following, where is some positive integer such that
(obtained from this source: memoryless property of geometric distribution)
b) Reciprocally, show that any random variable , having nonnegative integer values, whose distribution satisfies the property above has necessarily a geometric distribution.
I presently don't have an answer to this half of the question, so I could use a hand.