>>A much more interesting distribution than the uniform distribution is the exponential.
It has many more applications and can be used to estimate our discrete random variable,
the poisson. The exponential distribution describes a wide range of situations. It can
provide the probability of service times for customers, like how long it takes to fill
your car with gas, or have a haircut. It will calculate the probability for the times between
arrivals at a toll booth on a highway, or between accidents on the same road. The exponential
is the cousin of the poisson distribution, but the difference is important. First, the
poisson is a discrete random variable where time, a continuous variable, is artificially
broken into discrete pieces. We saw that the number of occurrences of an event in a given
time interval follows a poisson distribution. For example, the telephone just rang. How
long will it be until it rings again? We are measuring length of time of the interval:
exponential; not events during an interval: poisson. Classic poisson distribution questions
are: How many people will arrive at my checkout window in the next hour? Classic exponential
distribution questions are: How long will it be until the next person arrives? The formula
of the exponential is probability of x equal one divided by mu times e raised to the negative
x divided by mu. We see immediately the similarity between the exponential formula and the poisson
formula. Both probability density functions are based upon the relationship between time
and exponential growth or decay. The e in the formula is a constant, with the approximate
value of two point seven one eight two eight, and is based on the natural logarithmic exponential
growth formula. When people say that something has grown exponentially, this is what they’re
talking about. Although almost never do they really know what they’re saying. Growing
exponentially has almost become just an expression. An example of the exponential and poisson
will make clear the difference between the two. It will also show the interesting applications
that they have. Suppose that historically ten customers arrive at the checkout line
each hour, remember that this is still probability, so we have to be told these historic values.
We see this as a poisson probability problem. We can put this information into the poisson
probability density function and get a general formula that will calculate the probability
of any specific number of customers arriving in the next hour. F of x equals ten to the
x power e to the minus ten divided by x factorial. This formula is for any value of the random
variable we choose. And so the x is put into the formula as an example the probability
of fifteen people arriving at the checkout counter in the next hour would be probability
x equals fifteen, ten to the fifteenth power, e to the tenth power, all divided by fifteen
factorial. Here we have inserted fifteen and calculate the probability that in the next
hour fifteen people will arrive. The answer is point zero six one. Of course if we keep
the same historical facts that are interested in the service time a person spends at the
counter we would use the exponential distribution. The exponential probability function for any
value of x, the random variable, for this particular check out counter is f of x equals
one divided by point one, e raised to the minus x divided by point one equals ten e
raised to the minus ten x power. To calculate mu, the historical average service time, we
simply divide the number of people, ten, into the time period, one hour. And have mu equal
to point one. Historically people spend point one of an hour at the checkout counter. Or
six minutes. This explains the point one in the formula. If we wanted to know the probability
that a person will spend nine minutes or less checking out we use this formula. First we
convert to the same time units, which are parts of one hour. Nine minutes if point one
five of an hour. Next we note that we are asking for a range of values. This is always
the case for a continuous random variable. We write the probability question as the probability
of x equal to or less than x naught equal one minus ten times e raised to the ten x
power, okay? The probability that a customer will spend nine minutes or less checking out
is point seven seven six nine. We find this by looking at the formula probability x equal
to or less than point five equals one minus e raised to the minus ten times point one
five. That equals point seven seven six nine. We see that we have a high probability of
getting out in less than nine minutes and a tiny probability of having fifteen customers
arrive in the next hour. Now we know why we see those clocks at the drive-in windows.
They’re developing historical data to calculate the probabilities that we have just been doing.
These can be very helpful in making decisions on staffing, back office flow of work, any
number of other decisions that effect the business and their customers.