The median of a probability distribution can be defined as the number m such that upper p (upper x less than or equals m )equals upper p (upper x greater than or equals m ). find an expression for the median m of the exponential distribution.

tex]M=\beta ln(2)[/tex]

Step-by-step explanation:

Previous concepts

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).

Solution to the problem

For this case we can use the following Theorem:

"If X is a continuos random variable of the exponential distribution with parameter for some "

Then the median of X is

Proof

Let M the median for the random variable X.

From the definition for the exponential distribution we know the denisty function of X is given by:

Since we need the median we can put this equation:

If we evaluate the integral we got this:

And that's equal to:

And if we solve for M we got:

If we apply natural log on both sides we got:

And then

it would be 6xy and -16xy.