Stat 61 - Homework #3 (9/26/07; due 10/1/07)
1. Roll two standard dice. Let X = larger of the two numbers showing.
a. Construct a table showing both the probability mass function (pmf) pX and the cumulative density function (cdf) FX.
b. What is FX(7) ? (Note: “pmf” = what the text calls a “pdf” in the case of a discrete r.v.)
2. Let Y be a binomial random variable with parameters n = 6 and p = 1/2.
a. What is P ( Y = 3 ) ?
b. What is P ( Y > 3 ) ? [Can you do part b with zero extra work?]
3. Z is a random variable with the pmf
pZ (k) = C (1/10)k if k = 1, 2, 3, 4, …
(or zero if n has any other value)
(in particular, pZ(0) = 0)
…where C is a particular real number. What is C ?
4. A bar of Uranium contains n atoms. Each of them has (independent) probability p of decaying during a one-minute test. Let W be the random variable whose value is the number of atoms that decay during the one-minute test.
What is P ( W = 2 ) if…
a. n = 2 and p = 0.5
b. n = 10 and p = 0.1
c. n = 100 and p = 0.01
d. n = 1023 and p = 10-23 (approximate if you must)
5. A discrete random variable has a Poisson distribution with parameter m and is called a Poisson random variable with parameter m if its pmf is given by
p(k) = e-m ( mk / k! ) if k = 0, 1, 2, 3, …
or 0 for any other value of k.
Show that (as we would hope)
p(0) + p(1) + p(2) + … = 1.
(This is really a Math 25 problem. Think of a Taylor series.)
6. (from text) A continuous random variable X has the following cdf:
FX (a) = 0 if a ≤ 0
a2 if 0 ≤ a ≤ 1
1 if a ≥ 1.
a. What is P ( 1/2 ≤ X ≤ 3/4 ) ?
b. What is the probability density function (pdf) for X ?
c. Express the answer to (a) as an integral involving the pdf.