5. The geometric and exponential distributions share the lack of memory property, defined as follows: P(X>...

 

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5. The geometric and exponential distributions share the lack of memory property, defined as follows: P(X> x+yX >x) = P(X > y). (a) Let X be geometric, with probability function f(x) = p(1 - p)¹ for x = 0,1,2,... and parameter 0 < p < 1. It counts the number of failures before the first success in a sequence of independent Bernoulli trials. Show that the lack of memory property holds, and interpret this result. (b) Let X have an exponential distribution with pdf f(x) = Xe for x > 0 and rate parameter x > 0. Show that the lack of memory property holds. Interpret this result. Hint: Recall that P(A/B) = P(An B)/P(B). 5. The geometric and exponential distributions share the lack of memory property, defined as follows: P(X> x+yX >x) = P(X > y). (a) Let X be geometric, with probability function f(x) = p(1 - p)¹ for x = 0,1,2,... and parameter 0 < p < 1. It counts the number of failures before the first success in a sequence of independent Bernoulli trials. Show that the lack of memory property holds, and interpret this result. (b) Let X have an exponential distribution with pdf f(x) = Xe for x > 0 and rate parameter x > 0. Show that the lack of memory property holds. Interpret this result. Hint: Recall that P(A/B) = P(An B)/P(B).

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