Question

Suppose that the time between successive occurrences of an event follows an exponential distribution with mean number of occurrences per minute given by λ = 5. Assume that an event occurs. (A) Derive the probability that more than 2 minutes elapses before the occurrence of the next event. Derive the probability that more than 4 minutes elapses before the occurrence of the next event. (B) Use to previous results to show: Given that 2 minutes have already elapsed, what is the probability that a further 2 minutes elapse before the next occurrence?

Answer 1

a.

occurences per min = 5

P(T>2) = e^(-5*2) = 0.00005

P(T>4) = e^(-5*4) = 2.06115362*10^-9

b.

if 2 min have elapsed further 2 min required means total more than 4 min required given 2 min elapsed

required : P(T>4 | T>2) = P(T>4 and T>2) / P(T>2)

P(T>4 and T>2) : we can say if T>4 then it obviously will be greater than 2

therefore :

P(T>4 and T>2) = P(T>4)

now,

P(T>4 | T>2) = P(T>4) / P(T>2)

= e^(-5*4) / e^(-5*2)

= e^(-5^2)

P(T>4 | T>2) = 0.00005

(please UPVOTE)