Math 5010 - 1 Homework Assignments

MATH 5010 § 1	HOMEWORK ASSIGNMENTS	Spring 2009
A. Treibergs

Problems taken from the text, Elementary Probability Theory, 2nd. ed. by David Stirzaker, Cambridge University Press, 2003. You are responsible for all of the listed problems. Please hand in only the starred (^*) problems.

Please make sure your papers are SELF-CONTAINED. Copy or paraphrase the statement of the problem. Use English sentences to give at least a minimal explanation of your answer. State any theorems or formulas you use.

ATK students homework is due the Wednesday following the Friday due date for regular students. Please email a PDF of your work or send a FAX (801-581-4148) to the Math department. Be sure to put my name on your message.

Thanks to Prof. Ethier for kindly making his solutions(^†) available.

Page[Problems] Due Date Solns^*,^†

1. 48 [ 1-3, 4^*, 6^*, 7, 9^* ] Jan. 16 ^* ^†

2. 48 [ 10-11, 15^*, 19, 22^*, 23^* ] Jan. 23 ^* ^†

3. 76 [ 1^*, 3, 5^*, 6 - 8, 9^* ] Jan. 30 ^* ^†

4. 76 [ 11, 12^*, 15, 19^*, 20, 23^*, 27, 34, 35 ], Feb. 6 ^* ^†

108 [ 1, 2^*, 4, 6 ] ^†

5. 108 [ 9 - 11, 15, 17, 19 ], Feb. 13 ^†

Sample problems for the first midterm. ^*

FIRST MIDTERM EXAM Feb. 11 ^*

6. 151 [ 2 - 5, 11, 16 ], Feb. 20 ^* ^†

Please do the additional exercises:
1.^* Problem 1 with 13 sound, 3 rotten and
choose 4 randomly.
A.^* Roll four dice. X is the number of sixes.
Find D, f_X, F_X, E(X). Graph f_X, F_X.
B.^* Let D={1,2,3,...,n} and f_X(i)=ci for i in D.
Find c to make f_X a p.m.f. Find F_X, E(X).

7. 151 [ 18, 25^*, 27, 29 ], Feb. 27 ^* ^†

Please do the additional exercises:
A.^* Suppose X ∼ Geom(p). Find P(X is odd),
P(X is even), P(X > k), P(2 ≤ X ≤ 9 | X ≥ 4).
For 1 ≤ k ≤ n, find P(X = k | X ≤ n). Letting k
be a natural number and Y = g(X) where
g(x) = min(x,k), find pmf f_Y(y). Find E(1/X).
B.^* An unfair coin is tossed repeatedly. Suppose
that the events of getting a head on the ith toss
are independent and P(H)=p. Let X be the number
of tosses it takes to first get three heads.
Derive the formulas for E(X) and Var(X).

8. 151 [ 14^*, 30, 33, 35, 40, 42^*, 45 ], Mar. 6 ^* ^†

Please do the additional exercises:
A. Let X be a random variable with finite second
moment. Let μ = E(X) and σ² = Var(X). Let
Y = (X - μ)/σ. Show E(Y)=0 and Var(Y)=1.
B.^* Roll five dice. Let X be the smallest number
of the five dice. Find P(X ≥ x), f_X(x), E(X).
C.^* Let X ~ Poisson(λ). Find f_X( x | X is odd)
and E( X | X is odd).

9. 226 [ 1-3, 6, 9^*, 10, 12, 14, 15, 18, 21, 24, 25^*] Mar. 13 ^* ^†

Please do the additional exercise:
A.^* Two cards are selected at random without
replacement from a standard deck. Let X be the
number of kings and Y be the number of clubs.
Determine the joint pmf f(x,y). Find the marginal
pmfs. Are X and Y independent? Find cov(X,Y)
and ρ(X,Y).

10. 226 [ 31^*, 32, 33, 35, 36^*, 39] Mar. 27 ^* ^†

Please do the additional exercise:
A.^* N people arrive separately to a professional
dinner. Upon arrival, each person looks to see
if he or she has any friends among those present.
That person then either sits at the table of a
friend or at an unoccupied table if none of those
present is a friend. Assuming that each of the ^NC₂
pairs of people are, independently, friends with
probability p, find the expected number of occupied
tables. [cf Ross, Ch 7, Prob. 8.]

11. Sample problems for the second midterm. Apr. 3 ^* ^† ^‡

SECOND MIDTERM EXAM Apr. 1 ^*

12. 281 [ 1^*, 2, 3, 6, 10, 17 ] Apr. 10 ^* ^†

Please do the additional exercises:
A.^* Consider the coupon collecting problem with
three different types of coupons. And let T be the
number of boxes needed until you first possess all
three types. Find P(T=k), E(T) and Var(T) using a
probability generating function.
B.^* Flip a fair coin repeatedly until you get two
consecutive heads. This takes X flips. Derive the
formula for the the pgf. Use it to find the
expectation and variance of X.

13. 334 [ 1, 2, 4, 5, 6, 7, 8, 10, 11, 14^*, 22 ] Apr. 17 ^* ^†

Please do the additional exercises:
A.^* For positive numbers A and r, define the
function f(x)=cx^-r-1 if x ≥ A and f(x)=0 if x < A.
For what c is f a probability density function?
Show that this distribution posseses a finite nth
moment if and only if n<r. Find the mean and
variance in cases where they exist.
B.^* Let X have a χ² distribution with parameter n.
Show that Y = (X/n)^1/2 has a χ distribution with
parameter n, i.e., with the probability density
function f_Y(y)=2(n/2)^n/2y^n-1exp(-(n/2)y²)/Γ(n/2)
for y>0 and f_Y(y)=0 for y≤0.

14. 334 [ 3, 13, 15, 24, 25^*, 33, 35 ], 391 [ 7^* ] Apr. 24 ^* ^†

Please do the additional exercises:
A.^* Suppose that X₁,X₂,X₃,... are independent
random variables, all Poisson distributed with
parameter λ = 2 and Y_n = (S_n - E(S_n)) / σ(S_n)
where S_n = X₁ + X₂ +...+ X_n. Show that Y_n
converges to the standard normal variable
Z ∼ N(0,1) in distribution. (Don't quote CLT.)
Approximate P( 18 < S₁₀ < 20 ).
15. 391 [ 1 - 5, 13, 16, 19, 27 ] May 1 ^†

16. Sample problems for the final. Apr. 29 ^* ^†

FINAL EXAM May 1 ^*

	Page[Problems]	Due Date	Solns^*,^†
1.	48 [ 1-3, 4^, 6^, 7, 9^* ]	Jan. 16	^* ^†
2.	48 [ 10-11, 15^, 19, 22^, 23^* ]	Jan. 23	^* ^†
3.	76 [ 1^, 3, 5^, 6 - 8, 9^* ]	Jan. 30	^* ^†
4.	76 [ 11, 12^, 15, 19^, 20, 23^*, 27, 34, 35 ],	Feb. 6	^* ^†
	108 [ 1, 2^*, 4, 6 ]		^†
5.	108 [ 9 - 11, 15, 17, 19 ],	Feb. 13	^†
	Sample problems for the first midterm.		^*
	FIRST MIDTERM EXAM	Feb. 11	^*
6.	151 [ 2 - 5, 11, 16 ],	Feb. 20	^* ^†
	Please do the additional exercises: 1.^* Problem 1 with 13 sound, 3 rotten and choose 4 randomly. A.^* Roll four dice. X is the number of sixes. Find D, f_X, F_X, E(X). Graph f_X, F_X. B.^* Let D={1,2,3,...,n} and f_X(i)=ci for i in D. Find c to make f_X a p.m.f. Find F_X, E(X).
7.	151 [ 18, 25^*, 27, 29 ],	Feb. 27	^* ^†
	Please do the additional exercises: A.^* Suppose X ∼ Geom(p). Find P(X is odd), P(X is even), P(X > k), P(2 ≤ X ≤ 9 \| X ≥ 4). For 1 ≤ k ≤ n, find P(X = k \| X ≤ n). Letting k be a natural number and Y = g(X) where g(x) = min(x,k), find pmf f_Y(y). Find E(1/X). B.^* An unfair coin is tossed repeatedly. Suppose that the events of getting a head on the ith toss are independent and P(H)=p. Let X be the number of tosses it takes to first get three heads. Derive the formulas for E(X) and Var(X).
8.	151 [ 14^, 30, 33, 35, 40, 42^, 45 ],	Mar. 6	^* ^†
	Please do the additional exercises: A. Let X be a random variable with finite second moment. Let μ = E(X) and σ² = Var(X). Let Y = (X - μ)/σ. Show E(Y)=0 and Var(Y)=1. B.^* Roll five dice. Let X be the smallest number of the five dice. Find P(X ≥ x), f_X(x), E(X). C.^* Let X ~ Poisson(λ). Find f_X( x \| X is odd) and E( X \| X is odd).
9.	226 [ 1-3, 6, 9^, 10, 12, 14, 15, 18, 21, 24, 25^]	Mar. 13	^* ^†
	Please do the additional exercise: A.^* Two cards are selected at random without replacement from a standard deck. Let X be the number of kings and Y be the number of clubs. Determine the joint pmf f(x,y). Find the marginal pmfs. Are X and Y independent? Find cov(X,Y) and ρ(X,Y).
10.	226 [ 31^, 32, 33, 35, 36^, 39]	Mar. 27	^* ^†
	Please do the additional exercise: A.^* N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the ^NC₂ pairs of people are, independently, friends with probability p, find the expected number of occupied tables. [cf Ross, Ch 7, Prob. 8.]
11.	Sample problems for the second midterm.	Apr. 3	^* ^† ^‡
	SECOND MIDTERM EXAM	Apr. 1	^*
12.	281 [ 1^*, 2, 3, 6, 10, 17 ]	Apr. 10	^* ^†
	Please do the additional exercises: A.^* Consider the coupon collecting problem with three different types of coupons. And let T be the number of boxes needed until you first possess all three types. Find P(T=k), E(T) and Var(T) using a probability generating function. B.^* Flip a fair coin repeatedly until you get two consecutive heads. This takes X flips. Derive the formula for the the pgf. Use it to find the expectation and variance of X.
13.	334 [ 1, 2, 4, 5, 6, 7, 8, 10, 11, 14^*, 22 ]	Apr. 17	^* ^†
	Please do the additional exercises: A.^* For positive numbers A and r, define the function f(x)=cx^-r-1 if x ≥ A and f(x)=0 if x < A. For what c is f a probability density function? Show that this distribution posseses a finite nth moment if and only if n<r. Find the mean and variance in cases where they exist. B.^* Let X have a χ² distribution with parameter n. Show that Y = (X/n)^1/2 has a χ distribution with parameter n, i.e., with the probability density function f_Y(y)=2(n/2)^n/2y^n-1exp(-(n/2)y²)/Γ(n/2) for y>0 and f_Y(y)=0 for y≤0.
14.	334 [ 3, 13, 15, 24, 25^, 33, 35 ], 391 [ 7^ ]	Apr. 24	^* ^†
	Please do the additional exercises: A.^* Suppose that X₁,X₂,X₃,... are independent random variables, all Poisson distributed with parameter λ = 2 and Y_n = (S_n - E(S_n)) / σ(S_n) where S_n = X₁ + X₂ +...+ X_n. Show that Y_n converges to the standard normal variable Z ∼ N(0,1) in distribution. (Don't quote CLT.) Approximate P( 18 < S₁₀ < 20 ).
15.	391 [ 1 - 5, 13, 16, 19, 27 ]	May 1	^†
16.	Sample problems for the final.	Apr. 29	^* ^†
	FINAL EXAM	May 1	^*

Last updated 5 - 7 - 9.