Welcome to Excel 2013, Statistical Analysis Video
Number 42.
And if you want to download this Excel workbook or the PDF files
and go along with this video, click on the link
below the video.
Hey, we've got to talk about the exponential probability
distribution.
Hey, there is a chart of it.
It looks like it's real tall at the beginning,
but then it gets smaller and smaller as it gets further
out the x.
So if we're talking about times between arrival or times
to stand in line, the longer the time,
the smaller the probability.
Hey, exponential is useful in computing probabilities
for the time it takes to complete
a task or distance between similar occurrences
or successes--
so examples, time between arrivals at a car wash.
Or our example last video with Poisson,
we talked about number of arrivals
at a restaurant per one minute.
We could do time between arrivals at that restaurant.
And we'll look at that example--
time to take a test or distance between potholes on a road.
All right, it's related to the Poisson,
but now we're talking about time between arrivals
instead of number of arrivals per time period.
Now, let's look at our functions.
Here is our exponential density function.
There it is that calculates the height of the curve.
If we use EXPON.DIST, we put a 0 to plot our curve.
If we use the exponential cumulative probability
formula--
there it is in EXPON.DIST.
If we use a 1, it'll calculate cumulative probability from 0
to x.
Now we want to scroll down and talk
about the relationship between Poisson and exponential
distributions.
Now, down here, B, we'll start with B first.
If the arrivals follow a Poisson distribution,
the time between arrivals must follow
an exponential distribution.
Now we'll look at our Dick's Hamburger example.
That's the one we did last chapter, where we had 3.88
people arriving per minute.
That's the Poisson, counting people
over a particular interval that are arriving.
If we want to switch to exponential,
that means we'd take the inverse of that
for our exponential distribution.
And for our Dick's Hamburger restaurant,
that means we'd have 15 seconds between each arrival.
That would be our average arrival time.
Now, Poisson distribution provides
an appropriate description of number
of occurrences per interval, 3.88 people arriving
per minute.
Exponential distribution provides
an appropriate description of the length of the interval
between occurrences or arrival.
In our Dick's example, it will be
an average of 15 seconds between each arrival.
Now let's go over to Excel.
And we want to look at this very example.
I'm on the sheet Exponential Dick's Hamburger Example.
Now, if I scroll over here, here is our example
from last time, number of arrivals during one minute.
There is our Poisson function to calculate the chart.
Here is our chart.
And we've got, basically, calculating probabilities
of number of people arriving in any one-minute period.
But now in this example, we want to talk about arrivals
to Dick's Hamburger at 45th Avenue in Seattle
on Saturday noon lunch hour during a one-minute period
is 3.88 people arriving per one minute.
That's what we did for Poisson.
Now we're going to see the average time between arrivals
is 0.2577 minutes, or about 15 seconds between arrivals.
Now, because there is this relationship,
if we know the mean number of people arriving per one minute,
we'll take the inverse of that, and that'll
give us the average time between arrivals.
So I'm going to say equals 1 divided by this 3.88.
What it means is, if I take the inverse of is,
it means for every one minute, I'm
going to divide that by 3.88 arrivals--
Enter.
What it gives me is 0.2577 minutes per one arrival.
Now, I actually want to convert this to seconds.
This is in minutes.
Now we want seconds, so equals this times 60.
Now I like to round this.
And I want to use our rounding rule.
I want around to the actual second.
So I'm looking at the 4.
I need to round this to 15 seconds.
F2-- I don't want to do that by hand.
I want to use the Round function.
So Round will to any number of digits.
Hey, I'm going to the integer, so I put a 0,
close parentheses, and Enter.
That rounds to the integer, in our case, seconds.
So there it is, average time between arrivals.
Now we can ask the question, hey,
what's the probability that time between arrivals
will be 10 seconds or less?
If I'm looking at this on the chart, there is 10 minutes.
It's going to be all of this probability right
here, from 0 up to 10.
Remember, the way our function will work
is, it'll automatically go from the lowest
up to whatever x value we give it.
So I'm going to do equals EXPON.DIST.
Hey, there is the x--
comma.
And our lambda is going to be 1 divided
by our average time between arrivals.
And then comma--
1 gives us cumulative from 0 to whatever our x.
And so boom, there is the probability.
The probability that the time between arrivals
will be 10 seconds or less is 0.4866.
Now, what's the probability that the time between arrivals
will be 30 seconds or more?
Oh, here, if we look at this visually, it's above, right?
So I'm going to come here.
It's going to be our 1 minus if we do EXPON.DIST,
if we put in this x, that calculates everything
from 0 up to our 30, right?
So that's why we say 1 minus this, comma.
Our lambda is going to be 1 divided
by our average time between arrivals, comma,
1 for cumulative, close parentheses, and Control Enter.
There is the probability, so 0.315 that
the time between arrivals will be 30 seconds or more.
I plotted this down here.
And if you want to plot it, it's similar.
Notice EXPON.DIST-- now, I actually took this lambda here,
because I wanted this to plot, even though I
didn't have these values.
But that lambda right there is just 1 divided by that.
Notice the 0 in the cumulative.
That allows us to plot the chart.
All right, now we want to go look
at our next example, Exponential Disney Wait Time.
Now, for a particular Disney ride,
the mean time to wait in line is 22 minutes.
The distribution follows an exponential distribution
pattern.
So this is the average time to stand in line.
And we want to calculate the probability
that we'll stand in line 15 minutes or less.
Now, we could use this formula, 1 minus e.
And we can use exponent 1 in Excel
raised to the minus particular value divided by mu.
We're going to go ahead and try that here--
equals 1 minus exponent 1--
that is the method of getting e--
carrot minus, and then in parentheses,
simply particular x divided by our mu, close parentheses
and Enter.
So the probability is 0.49, almost 50%
that we could stand in line 15 minutes or less.
Now if we added this to the chart down here,
it would look like that, that whole area right there.
Now let's go ahead and use our EXPON.DIST, our particular x,
comma, 1 divided by our mean.
And cumulative is 1.
That means we're going from the 0
all the way up to our particular x, close parentheses,
and Enter-- which ever way you like it, the actual formula
or the exponent.
Now, if we want to calculate greater than
or equal to 25 minutes in line--
notice this formula is 1 minus.
So if we did 1 minus 1, it would change.
It would just be 0 there.
And it would change that minus to a plus.
So we can simply do the exponent equals [? expon ?] 1.
That just gives you my e.
If I enter it, it says 2.7182 repeating decimals.
That's that magic number.
And we're going to simply raise it
to minus, open parentheses, our particular x.
And we're divided by our mean time, close parentheses,
and Enter.
So the probability that we'll stand in line 25 minutes
or greater is 0.32.
If I add this to the chart down here, it's everything above.
Now, we could just use our EXPON.
And we'll use the same convention.
We're after probably on the upper end.
The EXPON.DIST will calculate from 0 up to our x,
so I'm going to do equals 1 minus EXPON.DIST,
our particular x, comma, 1 divided by our mean wait
time, which is lambda, comma, 1, close parentheses, Enter.
And we get the same exact thing.
Now we could do between.
Notice we're after this.
Well, we need to get rid of this and this.
All the area is 1, so I can simply say equals 1 minus,
in whichever order, get rid of that one by minusing
and then get rid of the other one by minusing, and Enter.
We could also flat out do our rule.
Anytime we're getting between, we
get bigger area minus smaller area, so equals EXPON.DIST.
Our big x is 25, comma, 1 divided by our mean, comma, 1.
That gives me all the area up to our 25--
F2.
And now I'm going to subtract EXPON.DIST the smaller x,
comma, 1 divided by--
and go up and get our mean wait time--
comma, 1, close parentheses, Control Enter.
And so we get exactly the same thing.
All right, in this video, we talked
about the exponential distribution,
great when you have mean wait times like for a Disney ride
or you have average time between arrivals like in our Dick's
Hamburger restaurant example.
All right, that's it for Chapter Six,
Continuous Probability Distribution.
Next chapter, we'll get to talk about the amazing central limit
theorem.
All right, we'll see you next video.