Mathematics 134 -- Intensive Calculus for Science 2
Discussion 3: Working with PDFs and CDFs
March 20, 2002
Background
Over the past days, we have introduced the ideas of probability
density functions (pdf's) and their cumulative distribution
functions (cdf's).
A pdf p(x) must satisfy p(x) >= 0 for all x,
and the total area between its graph and the x-axis must
equal 1. For a random variable x
with pdf p(x), the integral of p(x) from x = a
to x = b gives the probability that a <= x <= b.
The cumulative distribution function is an antiderivative P(t) of
p(x). The value P(t) gives the probability that
x <= t. Today, we want to practice using these ideas.
Discussion Questions
- A) Let x be a random variable representing the time
in minutes, starting from t = 0, that the next customer
walks into a store. Which of the following functions could be the
pdf for x? Which is the best choice for the pdf
in this situation? Explain how you are ruling out the ones you are
ruling out, and why you are keeping the ones you are keeping.
- p(x) = 1/5 for 1 <= x <= 6 (and 0 otherwise)
- p(x) = sin(x) for 0 <= x <= 2 Pi (and 0 otherwise)
- p(x) = 2e-2x for x >= 0 (and 0 otherwise)
- B) Consider a group of people who have received treatment for
an illness such as heart disease. The survival time, x is the number
of years a person lives after receiving treatment. If the group
of people is large enough, then we can model x as a
random variable with a density function
of the form p(x) = c e-cx for some
constant c
(for x >= 0 and p(x) = 0 for x < 0).
- If a randomly selected patient has a .3 (30%) chance
of surviving at most 2 years (or what is the same, a .7 (70%) chance
of surviving at least 2 years), what is the value of c
in the density function?
- What is practical meaning of the cdf P(t) in
this setting? Say what the value of P(t) tells
you in words. Plot P(t) for the pdf from part 1.
- C) In class we discussed the mean, or expected value
of a random variable, which is analogous to the center of mass.
The mean of the random variable x with pdf p(x)
is the integral of x p(x) from x = -infinity to x = infinity.
We can also discuss the median value, which the t-value
t = T such that P(T) = 1/2 (so exactly 1/2
of the population has t <= T, the median value -- and
hence 1/2 has t >= T as well).
- What are the median and the mean for the uniform pdf
p(x) = 1/5 for 1 <= x <= 6 (and 0 otherwise)?
- Are the median and mean the same for the exponential
pdf p(x) = 2e-2x for x >=0 (and 0 otherwise)?
Explain.
- D) The following table gives the cdf for incomes in the US in 1973.
| Income | 1000 | 4400 | 7800 | 12600 | 20000 | 50000
|
|---|
| P(t) | .01 | .10 | .25 | .50 | .75 | .99
|
|---|
- What fraction of the population made between $20000 and $50000?
- What was the median income?
- Estimate the mean income from this information (how?)
Assignment
Solutions will be due in class, Wednesday, April 3.
One set of solutions per group as usual.