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 ≤ 2) 

(d) P(3 ≤ X ≤ 5)

Answer 1

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

So,$F(x)=P(X<x)=1-e^{-\lambda x}=1-e^{-x},x>0$

a)

$E(X)=\frac{1}{\lambda}=\frac{1}{1}=1$

b)

Std. Dev.(X)

c)

$P(X\leq 2)=F(2)=1-e^{-2}=0.865$

d)

$P(3\leq X\leq 5)=F(5)-F(3)=[1-e^{-5}]-[1-e^{-3}]=0.043$