Stat 61 - Homework #5 (10/22/07, replaces draft 10/12/07; due 10/29/07)
1. Let X be a random variable with E(X) = 100 and Var(X) = 15. What are…
a. E ( X2 ) (Not 10000)
b. E ( 3X + 10 )
c. E (-X)
d. Standard deviation of –X ?
2. Let X be the random variable with pdf
fX ( x ) = 2x if 0 ≤ x ≤ 1
a. What is the cdf, FX(a) ?
b. Let Y be another random variable defined by Y = X2. What is its cdf,
FY (b) ?
( This is possible because if you know P( X ≤ a ) for every a, you can get
P( Y ≤ b ) for any b.)
c. What is the pdf, fY ?
d. Compute E(Y) directly from FY. (Use the secret formula.) (See website if necessary)
e. Compute E(Y) from fY.
f. Compute E(Y) = E(X2) directly from fX. (Part f is the Lazy Statistician’s
Rule. If your answers to d-e-f don’t agree, panic.)
3. (Sum of random variables) Let X and Y be independent random variables, with these pmf’s:
pX(n) = (1/3) for n = 1, 2, 3; else 0
pY(n) = (1/2) for n = 1, 2; else 0
Let Z = X + Y. Using the formula
construct the pmf for Z.
4. Now let X and Y be independent Poisson random variables, with means mX and mY respectively. Let Z = X + Y. Using the formula from problem 3, derive the pmf for Z.
(Hint: The binomial formula is . )
5. (The mother of all Poisson examples) Do problem 4.2.10, page 287. Either actually read the problem and do what it says, or do parts a and b below, which amount to the same thing. Specifically, given the following distribution…
k Observed number of
corps-years in which
k fatalities occurred
[This is a frequency table describing 200 annual reports from pre-WWI Prussian cavalry corps, each giving a number of soldiers killed by being kicked by horses.]
It is suggested that these values represent 200 independent draws from a Poisson distribution.
a. What is a reasonable estimate for the mean of the distribution?
b. What would the entries in the right-hand column be if the 200 draws were exactly distributed according to this Poisson distribution? (The entries would not be integers.)
(In November we’ll address this question: Comparing the original data to the theoretical values in part b, is it plausible that these really are draws from a Poisson distribution? Or should we discard that theory?)
6. (Editing a Poisson distribution) At a certain boardwalk attraction, customers appear according to a Poisson process at a rate of l = 15 customers/hour. So, if X is the number of customers appearing between noon and 1pm, X has a Poisson distribution with mean 15.
Assume that each customer wins a prize with (independent) probability 1/5.
Let Y be the number of customers winning prizes between noon and 1pm.
(problem 6 continues on next pageà)
(Problem 6 continued)
One way to understand Y is that if the value of X is given, then Y is binomial, with parameters n = [value of X] and p = 1/5. This reasoning gives us:
P ( Y = k ) =
(The second part of the summand is the probability that Y=k given that X=n.)
a. Simplify this formula, to show that Y is itself a Poisson distribution with mean 3.
(Or, if after getting off to a good start you find this problem too annoying, do
part b instead.)
b. Find a much simpler explanation of why Y should be Poisson with mean 3.
7. (Waiting times) Buses arrive according to a Poisson process with mean (1/10) min-1. Let W be the waiting time from time t = 0 till the arrival of the second bus.
a. What is E(W) ?
b. Can you construct a pdf or cdf for W ?
( Good start: P(W ≤ t) = 1 – P(exactly 0 buses or exactly 1 bus between times 0 and t). Another approach: W is the sum of two independent one-bus waiting times.)
8. (The first boring normal tables problem) If Z is a standard normal variable, what are…
a. P ( -1.0 ≤ Z ≤ +1.0 )
b. P ( -2.0 ≤ Z ≤ +2.0 )
c. P ( -3.0 ≤ Z ≤ +3.0 )
9. (The second boring normal tables problem) Let Z be a standard normal variable.
a. If P ( -a ≤ Z ≤ +a ) = 0.95, what is a ?
b. If P ( -b ≤ Z ≤ +b ) = 0.99, what is b ?
10. (The third boring normal tables problem)
a. If X is normal with mean 500 and standard deviation 110,
what is P ( X ≥ 800 ) ?
b. If X is normal with mean 500 and standard deviation 120,
what is P ( X ≥ 800 ) ?