Question

Let U=min(X1, X2), the minimum of two independent exponentials. What is the distribution of U?

Answer 1

GIVEN:

Let be the minimum of two independent exponential random variables.

TO FIND:

Distribution of U

SOLUTION:

Suppose that and are independent exponential random variables with mean,

and

Let

In order to find the distribution of U, first of all, since and this means that too.

So the density function of U is 0 for . Now let’s focus on the density function of U, for .

Easier to find the CDF of U. In fact, even easier to find the complement of the CDF of U.

If we want P(U > a) for some a > 0, we just want

   {Since and are independent.}

So the cumulative distribution function of U is given by,

So the cumulative distribution function of U has exactly the form of the cumulative distribution function of an exponential random variable, so U itself is exponential with

Also the density function for and otherwise.

Thus is an exponential random variable with parameter with mean .