Math 2015
Lesson 19
Density Functions and Probability
We begin a discussion of density functions, which are used to describe what proportion of a
population has a certain characteristic. Density functions will lead us into a discussion of
probability. We use density functions to represent information about the distribution of
various quantities through the population.
What is a Density Function?
You can think of a density function as a function that exists only to be integrated.
Suppose we want to measure some characteristic of a population, such as the height of
women in the U.S., the age of a trees in a valley, or the scores on the last calculus test. If
p(x) is a density function for the characteristic x, then
By itself, p(x) doesn’t mean much. But it does tell us how “densely” the population is
distributed around certain values. The examples we have given are all for what we call
continuous variables: they could take on a whole range of values, including decimals.
A Property and an Example
Let’s look at an example. Suppose the age of the trees in a forest is given by the density
function plotted below:
h
25
50
75
Let the peak’s height h on the y-axis be an unknown value for the moment. Now note that
there is no area under the curve for x > 75 and none for x < 0. According to our density
function, all the trees are between 0 and 75 years of age. Since all the trees have to be
accounted for, we must have
That is, the fraction of trees that are between 0 and 75 years old must be 1.
This will be a general property for density functions, which we will write in the following
way:
∫
∞
−∞
p(x)dx = 1
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Lesson 19
Recall that an improper integral like this means we are going to take the integral over larger
and larger intervals and see what the limit is. In our example with the trees, our density
function p(x) = 0 for x > 75 or x < 0. So if we know
•
∫
•
∫
100
−100
∫
75
0
p(x) dx = 1 , what is
p(x) dx ?
1000
−1000
p(x)dx ?
This pattern clearly continues. So we say that
∫
∞
−∞
p(x)dx = 1.
(We can also think of this improper integral as saying that we integrate everything. Since all
of our trees are between 0 and 75, “minus infinity to infinity” just becomes 0 to 75.
Whenever you integrate a density function everywhere, you must get 1.)
Now consider the following based on the density function given above:
1) What fraction of the trees are 50 years old or younger?
2) What fraction are older than 50 years?
3) What is the value of h? (Hint: What proportion of trees are older or younger than 50?)
A funny thing about density functions...
Suppose we wanted to use the density function above to determine how many trees are
exactly 27.32 years old. Then we would need to calculate
.
We won’t get any different answer if we ask how many trees are exactly 50 years old, or
exactly 14.2 years old, or any other age! What’s going on?
Density function are really only useful for asking about ranges. If we try to ask about a
specific value, we just get “0” as our answer. (This actually makes some sense, if we think
about our example with the trees: If our function said there were some number of trees for
every real number, then since there are infinitely many real numbers, we would need to have
a lot of trees.)
It is also possible to use a density function to analyze data which is discrete, meaning it
comes in separate pieces, rather than ranges of values. If you flip a coin 50 times, the
number of times it lands heads up is a discrete variable; it can only be numbers like 7, 8, or 9,
and never 7.32. It turns out that you can use a density function to measure such a variable,
but you must adjust your methods.
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Lesson 19
Some Properties
Here are a few useful things to know about density functions:
∫
∞
p(x)dx = 1.
•
We already know that
•
p(x) ≥ 0. If p were ever negative, we could have some interval over which we
could get a negative answer when we integrated.
Often (but not always), p(x) > 0 only over some finite interval.
•
−∞
Another Example
Here’s another example:
Example:
A particular piece of equipment needed in a factory may last up to 10 years.
If t is in years, the density function for the life of the machine is as shown
below:
h
0.01
5
10
a) Are the machines more likely to break in the first or tenth year?
Are they more likely to break in the first or second year?
b) What fraction lasts between 5–7 years? First find h.
Between 3–6 years?
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Lesson 19
Probability
In the last example, we used an area argument to say that the fraction of machines that lasted
between 5 and 7 years was 0.38 or 38%.
We could also interpret this answer as a probability: If we select a machine at random, it has
probability 0.38 or 38% of failing between year 5 and year 7.
Example:
According to the density function in the previous example, what is the
probability of a machine lasting more than five years? Less than five years?
In general, if x is a variable we want to measure a probability for, and p is a so-called
probability density function, then the probability of the value of x being between a and b is
given by
Example:
Suppose we have a spinner (as in a game) that can point to any angle from 0
to 360 degrees. Suppose that the spinner is fair; that is, the spinner is no more
likely to stop in one place than in any other. (So for example, the odds of
landing between 0 and 10 degrees are the same as the odds of landing between
83 and 93 degrees.)
What is the density function p(x) for the angle that the spinner lands on?
According to this density function, what is the probability that the spinner lands
• at an angle less than 10 degrees?
• at an angle less than 90 degrees?
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Math 2015
Lesson 19
• at an angle less than 360 degrees?
Suppose the spinner is somewhat sticky, so that the arrow is much more likely to stop near 90
degrees than anywhere else, sketch a reasonable density function for such a spinner:
Summary
Today, we have
∫
∞
p(x)dx = 1.)
•
Defined a density function and examined its properties (such as
•
Used a density function to determine (through integration) the fraction of a population
which has a particular range for a variable.
•
Interpreted a density function as a probability. The probability of a value being
between a and b can be represented by
∫
b
a
−∞
p(x)dx .
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