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 ? = 1, (which is identical to a standard gamma distribution with? = 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(3 ? X ? 5)

Answer 1

a) 1

b) 1

c)
`P(X\leq 4)=0.982`

`P(X\geq 4)=0.018`

d)
`P(3\leq X\leq 5)=0.043`

Step-by-step explanation:

An exponential distribution is a probability distribution of the time between events which occur continuously and independently at a constant average rate
`\lambda`. The probability function is

`f(X) = \lambda e^(-\lambda X) ; x\geq 0`

a) The expected value of an exponentially distributed variable X with rate
`\lambda` (in this case 1) is given by

`E[X]=1/\lambda`

E[X] = 1/1 = 1

b) The variance of an exponentially distributed variable X with rate
`\lambda` is

`Var[X]=1/\lambda^2`

so the standard deviation is

`\sigma=\sqrt{(1)/(\lambda^2)} = 1/\lambda`

`\sigma= 1/1 = 1`

c) the cumulative distribution function is

`F(X) = 1-e^(-\lambda X) ; x\geq 0`

with that function we can calculate the probability of time between events be lower or equal than determined number.

So
`P(X\leq4) = 1 - e^(-1*4)=0.982`

If you need to calcuate the probability of time between events be higher than determined number you can do the following operation

`P(X \geq 4) = 1 - P(X\leq4) = 1-0.982=0.018`

d)
`P(3\leq X\leq 5)` we need to make a substraction

`P(3\leq X\leq 5)= P(X \leq 5 ) - P(X \leq 3 )`

`P(3\leq X\leq 5)= 1-e^(-1*3) - (1-e^(-1*5)) = e^(-1*5) - e^(-1*3)=0.050-0.007= 0.043`