Welcome to a lesson on the memorylessness
of the exponential distribution.
Suppose that an average of 30 customers per hour arrive
at a store and the time
between arrivals is exponentially distributed.
On average, two minutes elapsed between successive visits.
Suppose that seven minutes have elapsed
since the last customer arrived.
Since an unusually long amount of time has now elapsed,
it would seem to be more likely for a customer to arrive
within the next minute.
With the exponential distribution though,
this is not the case.
The additional time spent waiting
for the next customer does not depend
on how much time has already elapsed
since the last customer.
This is referred to as the memoryless property.
Specifically, the memoryless property states
that the probability that X is greater than r plus t,
given X is greater than r, is equal to the probability
that X is greater than t.
And this is true for all r and t,
greater than or equal to zero.
For example, if five minutes have elapsed
since the last customer arrived,
then the probability that more than one minute will elapse
before the next customer arrives is computed
by using r equals five, and t equals one
in the equation above, which gives us the probability
that X is greater than five plus one,
given X is greater than five,
is equal to the probability that X is greater than one.
Again, where r is five and t is one.
Let's model this probability using area
under the probability density function graphed below.
Well, X equals one is here.
Since we want the probability that X is greater than one,
we want the area under the curve to the right,
but remember the cumulative distribution function,
big F of x gives us the area to the left, not the right,
and therefore to determine this probability
we need to find the area under the entire curve,
which is one, and then subtract the area to the left of one,
which is this green area,
which leaves us with the desired area in red
to the right of one, which I've already set up here,
which gives us approximately 0.6065, or 60.65%.
Notice how this probability is the same
as the probability of waiting more than one minute
for a customer to arrive after the previous arrival.
So the five minutes that elapsed does not affect
the probability the next customer will arrive
in more than one minute.
The exponential distribution is often used
to model the longevity of an electrical
or a mechanical device.
For example, the lifetime of a certain computer part
has the exponential distribution, with a mean of 10 years.
And because the mean is 10 years,
the decay parameter, m, is reciprocal of 10
or one divided by 10, which is 1/10, or 0.1.
The memoryless property says that knowledge
of what has occurred in the past has no effect
on future probabilities.
In this case, it means that an old part is not more likely
to break down at any particular time than a brand new part.
In other words, the part remains
as good as new until it suddenly breaks.
For example, if the part has already lasted 10 years,
then the probability that it lasts
another seven years is using the formula shown here
where r is equal to 10 and t is equal to seven.
We have the probability that X is greater than 17,
given X is greater than 10,
is simply equal to the probability
that X is greater than seven,
which comes out to approximately 0.4966.
And again, the memoryless property states
that the part already lasting 10 years has no effect
on whether it will last more than seven additional years.
I hope you found this helpful.