Question

# Suppose that the time between successive occurrences of an event follows an exponential distribution with mean...

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?

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

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