Acontinuous random variable, x, whose probability density function is given by f(x) = ( λe−λx , if x ≥ 0 0, otherwise is said to be an exponential random variable with parameter λ denoted by x ∼ exp(λ) where λ > 0.

(a) find the c. d.f of x.
(b) suppose that the length of a phone call in minutes is an exponential random variable with parameter λ = 1 10 . if someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait between 10 and 20 minutes

a)

b)

Explanation:

Previous concepts

The cumulative distribution function (CDF) F(x),"describes the probability that a random variableX with a given probability distribution will be found at a value less than or equal to x".

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution".

Part a

Let X the random variable of interest. We know on this case that

And we know the probability denisty function for x given by:

In order to find the cdf we need to do the following integral:

Part b

Assuming that , then the density function is given by:

And for this case we want this probability:

And evaluating the integral we got:

answer:

clock

explanation:

always running (on) sing each hour (chime) and the clock face (front)