Proving the memoryless property of the exponential distribution

DanielJackins

Given that a random variable X follows an Exponential Distribution with paramater β, how would you prove the memoryless property?

That is, that P(X ≤ a + b|X > a) = P(X ≤ b)

The only step I can really think of doing is rewriting the left side as [P((X ≤ a + b) ^ (X > a))]/P(X > a). Where can I go from there?

haruspex

P[(X ≤ a + b) ^ (X > a)] = P[X ≤ a + b] - P[X ≤ a] , right?

DanielJackins

Thanks. Using that I was able to prove it. But why is what you said true?

haruspex

X has three disjoint ranges, <a, (a,a+b), >a+b.
P[(X ≤ a + b) ^ (X > a)] is the probability X is in the middle range.
P[(X ≤ a + b)] is the probability X is in the first or middle range.
P[X ≤ a] is the probability X is in the first range.