In this example, we will find a percentile
of an exponential distribution.
For some background on the problem,
the random variable X is equal to the amount of time
in minutes a postal clerk spends with his or her customers.
The time has known to have an exponential distribution
with the average amount of time equal to four minutes,
and therefore mu, or the mean, is equal to four minutes,
m, the decay parameter is equal to one divided by mu
which in this case is 1/4, or 0.25.
And therefore the probability density function
is f of x equals 0.25 times e,
raised to the power of negative 0.25x,
which gives us the exponential distribution,
which is shown here.
And the cumulative distribution function
is big F of x equals one minus e,
raised to the power of negative 0.25x,
which gives the area to the left of x.
So now going to our problem, we want to answer the question,
"70% of all customers are finished within how long?"
Which would be the 70th percentile.
So we're looking for some value of x, let's call it k.
There will be, let's say approximately here,
where 70% of the customer times are less than
or equal to k and therefore the area
to the left of k is equal to 0.7.
So we can say the probability that x is less than k
is equal to 0.7.
And since this is the area to the left of k,
it must equal one minus e, raised to the power
of negative 0.25k,
which gives us the equation one minus e,
raised to the power of negative 0.25k, must equal 0.7.
And now if we solve this equation for k,
we can determine the 70th percentile.
So for the first step, let's subtract one on both sides,
which gives us negative e raised to the power
of negative 0.25k, is equal to 0.7 minus one,
which gives us negative e to the power
of negative 0.25k equals negative 0.3.
Let's multiply both sides of the equation by negative one,
which gives us e to the power of negative 0.25k equals 0.3.
And now to solve this exponential equation with base e,
we will rewrite the equation as a log equation
using the definition of a logarithm shown here,
where e to the power of x equals y is equivalent
to natural log y equals x,
where x is the exponent, e is the base.
Remember natural log is base e and y is a number.
So the equivalent log equation is natural log
because we have log base e.
A logarithm is an exponent,
and therefore the log is equal to negative 0.25k.
And the exponential expression is equal to 0.3,
which is the number part of the logarithm.
Notice now, to solve for k,
we divide both sides by negative 0.25.
Simplifying.
We have k is equal to the quotient on the left,
which we'll get a decimal approximation for.
We have natural log 0.3,
divided by negative 0.25.
To four decimal places we have approximately 4.8159,
which means that 70% of customers are finished
within 4.8159 minutes,
which also means 4.8159 is the 70th percentile
of finishing times, meaning 70% of the finishing times
are less than or equal to 4.8159 minutes.
Now, I also want to mention,
there is a quick plumbing that we can use
to find the percentile k given by this formula here.
And let's go and just verify we get the same value.
We would have k the 70th percentile is equal
to natural log of one minus the area to the left,
which is the 0.7
divided by the opposite of m, which is negative 0.25.
Simplifying.
We do get the same expression for k
of natural log 0.3 divided by a negative 0.25,
which we already saw does give us approximately 4.8159.
So if your instructor lets you use the shortcut formula,
it's obviously much faster
than solving the equation algebraically.
I hope you found this helpful.