Question

Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with λ = 1, (which is identical to a standard gamma distribution with α = 1), compute the following. (If necessary, round your answer to three decimal places.) 

(a) The expected time between two successive arrivals 

(b) The standard deviation of the time between successive arrivals 

(c) P(X ≤ 4) 

(d) P(2 ≤X≤5)

Answer 1

We are given the distribution here as:

$X \sim exp(\lambda = 1)$

a) The expected time between two successive arrivals for an exponential distribution is equal to its parameter which is 1 here. Therefore 1 is the required expected time here.

b) For an exponential distribution, the standard deviation is equal to its mean, therefore the standard deviation is also here equal to 1.

c) The probability here is computed as:

$P(X \leq 4) = \int_{0}^{4}e^{-x} \ dx = 1 - e^{-4} = 0.9817$

Therefore 0.982 is the required probability here.

d) The probability here is computed as:

$P(2 \leq X \leq 5) = \int_{2}^{5}e^{-x} \ dx = e^{-2} - e^{-5} = 0.129$

Therefore 0.129 is the required probability here.