3-The Exponential random variables 

A continuous random variable whose probability density function is given, for some λ > 0, by 

is said to be an exponential random variable (or, more simply, is said to be

exponentially  distributed)  with  parameter λ. 

The  cumulative distribution function F(a)of an exponential random variable is given by

Let X be an exponential random variable with parameter λ.

Calculate (a) E(X) and (b) Var(X).

(a)Since the density function is given by 

we obtain, for n > 0,

Letting n = 1 and then n = 2 gives 

(b)Hence, 

Thus, the mean of the exponential is the reciprocal of its parameter λ, and the variance is the mean squared. 

Suppose that the length of a phone call in minutes is an exponential

random variable with parameter  If someone arrives immediately ahead of you at a public telephone booth,

find the probability that you will have to wait

(a) more than 10 minutes;

(b) between 10 and 20 minutes.

Let X denote the length of the call made by the person in the booth.

Then the desired probabilities are 

(a)

(b)

The arrival rate of cars at a gas station is 𝜆 = 40 customers per hour.

(This is equivalent to saying that the interarrival times are exponentially distributed with rate 40 per hour.)

(a)What is the probability of having no arrivals in a 5-minute interval? 

(b)What are the mean and variance of the number, N, of arrivals in 5 minutes? 

(c)What  is  the  probability  for  having  3 arrivals  in a  5  minute interval? 

(a) 

(b) The variable N has a Poisson distribution with parameter 

𝐸(𝑁)=3.333         𝑎𝑛𝑑        𝑉𝑎𝑟(𝑁)=3.333.

(c)