Question

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.

Answer 1

Answer:

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 $\beta$ for some $\beta \in R >0$"

Then the median of X is $\beta ln (2)$

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:

$f_X (x) = \frac{1}{\beta} e^{-\frac{x}{\beta}}$

Since we need the median we can put this equation:

$P(X$

If we evaluate the integral we got this:

$\frac{1}{\beta} \int_0^M e^{- \frac{x}{\beta}}dx =\frac{1}{\beta} [-\beta e^{-\frac{x}{\beta}}] \Big|_0^M$

And that's equal to:

$1/2 = 1 -e^{- \frac{M}{\beta}}$

And if we solve for M we got:

$1-e^{- \frac{M}{\beta}} = \frac{1}{2}$

$e^{- \frac{M}{\beta}}=\frac{1}{2}$

If we apply natural log on both sides we got:

$-\frac{M}{\beta}=ln(1/2)$

And then $M=\beta ln(2)$