EXPONENTIAL DISTRIBUTION ASSIGNMENT HELP

Assume that X is a continuous random variable. Apart from that, let both X and λ be elements in the set of real numbers. They should only be contained in the set of positive real numbers. Then we say that X has an exponential distribution usually denoted as X ~ Exp(m) if its probability density function (pdf) is given as:

Where;
λ is a parameter of the exponential distribution
m is the average waiting period for the subsequent event relapse i.e. λ = 1/m

This parameter is usually called the rate parameter. It then follows that an exponential random variable X is a random variable that has an exponential distribution.
The cumulative distribution function (cdf) of an exponential random variable X is given as:

On the other hand, the moment generating function of an exponential random variable is given as:
M(t)=λ/(λ-t)
Where t<λ

Similarly, the characteristic function of any exponential random variable is given as:
φ(t)=λ/(λ-it)

Properties

Mean = µ = E(X) = 1/λ
Variance = Var(X) = 1/λ²

In an exponential distribution, any additional time spent waiting for the subsequent event to occur is independent of the amount of time that has elapsed since the last event. This property of an exponential distribution is called the memory less property and is expressed as:

P(X > x + k | X > x) = P(X > k)

It is worth mentioning that exponential distribution is a special case of gamma distribution with α=1.