# Using the probability density function to find the cumulative distribution function of an exponentially distributed random variable.

You are told that the continuous random variable X is exponentially distributed with parameter a (a > 0). A standard result then says that the probability density function of X is (see attachment)

f(x) = aexp(-ax) for x > 0.

Use this to prove that the corresponding cumulative distribution function F(x) (sometimes referred to as the distribtuion function) is

F(x) = 1 - exp(-ax).

Hence find the probability that X is greater than 5, if a=0.5

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#### Solution Preview

Denote the cumulative distribution function of a continuous random variable X by F(x). This is related to the probability density function f(x) via the relation (see attachment)

F(x)=P(X < x) = ...

#### Solution Summary

You are told that the continuous random variable X is exponentially distributed with parameter a (a > 0). A standard result then says that the probability density function of X is (see attachment)

f(x) = aexp(-ax) for x > 0.

Use this to prove that the corresponding cumulative distribution function F(x) (sometimes referred to as the distribtuion function) is

F(x) = 1 - exp(-ax).

Hence find the probability that X is greater than 5, if a=0.5