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) p_X and the cumulative density function (cdf) F_X.

b. What is F_X(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

p_Z (k) = C (1/10)^k if k = 1, 2, 3, 4, …

(or zero if n has any other value)

(in particular, p_Z(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 ( m^k / 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:

F_X (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.

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