5. Lack-of-memory property. If X has the exponential distribution, show that P

????

X > u + v X > u

= P(X > v) for u, v > 0.

This is called the ‘lack of memory’ property, since it says that, if we are given that X > u, then the distribution of X − u is the same as the original distribution of X. Show that if Y is a positive, continuous random variable with the lack-of-memory property above, then Y has the exponential distribution.