Welcome to a lesson
on the exponential distribution.
The exponential distribution is often concerned
with the amount of time until some specific event occurs.
For example, the amount of time, beginning now,
until an earthquake occurs has an exponential distribution.
Other examples include the length, in minutes,
of long-distance business telephone calls
and the amount of time, in months, a car battery lasts.
It can be shown too that the value of the change
that you have in your pocket or purse
approximately follows an exponential distribution.
Values for an exponential random variable occur
in the following way.
There are fewer large values and more smaller values.
For example, the amount of money customers spend
in one trip to the supermarket
follows an exponential distribution.
There are more people who spend small amounts of money
and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations
of product reliability,
or the length of time a product lasts.
As an example, let's let the random variable x equal
the amount of time, in minutes, a postal clerk spends
with his or her customer.
The time is known to have an exponential distribution
with the average amount of time equal to four minutes,
meaning the mean, or mu, is equal to four minutes.
x is a continuous random variable,
since time is measured.
It is given that mu, or the mean, is four minutes.
To do any calculations, we must know m,
the decay parameter, which is equal to the reciprocal
of the mean, or one divided by mu.
In this case, m is equal to one divided by four,
which is equal to 1/4, or 0.25.
The standard deviation, sigma, is the same as the mean.
Exponential distribution notation is shown here,
where again, m is a decay parameter,
which in our case, again, is 0.25.
The probability density function is f of x equals m times e
raised to the power of negative mx,
where e is an irrational number approximately 2.718.
In this example, since m is equal to 0.25,
the probability density function is f of x equals 0.25
times e raised to the power of negative 0.25 x,
where x is at least zero, and again, m is 0.25.
The cumulative distribution function
gives the area to the left
under the exponential distribution,
which is given by a big F of x equals one minus e
raised to the power of negative mx, or in our case,
big F of x is equal to one minus e
raised to the power of negative 0.25 x.
Let's look at the graph of the probability function,
which is here on the coordinate plane.
Notice how there are more smaller values
and less larger values in the exponential distribution.
that a postal clerk spends less than five minutes
with a random customer, which is the probability
that x is less than five.
To find this probability,
we will have to use the cumulative distribution function,
which is big F of x equals one minus e
raised to the power of negative mx,
and this gives the area to the left of the value of x.
So for the probability that x is less than five,
going back to the graph, here's where x equals five.
Because we're looking for the probability
that x is less than five,
we want to find the area under the curve
to the left of five, which is this area here,
which is on the left.
And therefore, we simply substitute five
into the cumulative distribution function
to find this probability.
So the probability is equal to one minus e
raised to the power of negative 0.25 times five.
Going back to the calculator.
We have one minus second natural log for e
raised to the power of negative 0.25 times five.
Enter.
To four decimal places, we have approximately 0.7135,
which is 71.35%.
For number three, we're asked to find the probability
that a postal clerk spends more than six minutes
with a random customer, which is the probability
that x is greater than six.
To find this probability, we need to find the area
under the probability density function.
So here is x equals six.
The area on the right, this area here,
is equal to the probability that x is greater than six.
But remember, the cumulative distribution function
gives the area to the left, not the right, and therefore,
to find this area, we will have to find the area
under the entire curve, which we know is equal to one,
this area here, and then subtract the area
to the left of six, which is this area,
which will leave is with the desired area,
which is the area to the right of six, this area here.
So the probability that x is greater than six
is equal to all the area under the curve,
which is equal to one minus the area under the curve
to the left of six,
which is equal to the quantity one minus e
to the power of negative 0.25 times six.
So now we'll go to the calculator.
We could simplify this expression,
but we'll go ahead and type it in just like it looks.
One minus open parenthesis,
one minus second natural log for e
to the power of negative 0.25 times six.
Right arrow, close parenthesis, Enter.
To four decimal places, we have approximately 0.2231,
which is equal to 22.31%.
For number four, we want to find the probability
that a postal clerk spends four to eight minutes
with a randomly selected customer,
which is the probability that x is greater than four
and less than eighth.
Let's go ahead and shade the area under the curve
between four and eight.
This area is equal to the probability that we want,
but again, we need to use
the cumulative distribution function,
which only gives the area to the left.
So to find the area between four and eight,
we will first find the area to the left of eight,
which is this area here,
and then subtract the area to the left of four,
which will leave us with the desired area,
which is the area between four and eight.
So this is equal to, again, the area to the left of eight,
which is the quantity one minus e
raised to the power of negative 0.25 times eight.
And then minus the area to the left of four,
which is equal to one minus e
raised to the power of negative 0.25 times four.
Now we'll go back to the calculator one last time.
So we have open parenthesis
one minus second natural log for e
to the power of negative 0.25 times eight,
right arrow, close parenthesis,
minus, open parenthesis, one minus second natural log for e
to the power of negative 0.25 times four,
right arrow, close parenthesis, and Enter.
To four decimal places,
we have approximately 0.2325,
which is equal to 23.25%
I hope you've found this helpful.