Problem Set 2. Counting
1. Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each
dinner being at one of his favorite ten restaurants.
i How many possibilities are there for Freds schedule of dinners for that w
Problem Set 11. Discrete 2D Distributions
1. Suppose a fair 6-sided die is rolled twice, and the RVs (X1 , X2 ) represent the values of the first
and second rolls. Let X = X1 + X2 be their sum and Y = X1 X2 be their difference.
i. Find the joint PMF of X,
STAB52 Week 7 Quiz
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1.
Let
STAB52 - Introduction to Probability Quiz 4 (week 9)
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1. (5 pts) Let Z Z(0, 1) be a stand
STAB52 - Introduction to Probability Quiz 4 (week 9)
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1. (5 pts) Let Z Z(0, 1) be a stand
STAB52 - Introduction to Probability Quiz 4 (week 9)
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STAB52 WEEK 5 QUIZ
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STAB52 Week 7 quiz
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1. For X ~ Geom(p), find E(2x) given that it is finite. Clearly state what value of p are for which
it is fini
Problem Set 9. 1D Change of Variables
1. Let X Poisson() and Y be the indicator of X being odd. Find the PMF of Y .
(Hint: Find P (Y = 0) P (Y = 1) by writing P (Y = 0) and P (Y = 1) as series and then using
the fact that (1)k is 1 if k is even and 1 if k
Problem Set 10. Expectations
1. Two teams reach a best-of-seven playoff, i.e. the first team to win 4 games become champions
and no more games are played beyond that. Assume both teams have the same dynamic, so
that each team has 1/2 probability of winnin
Problem Set 19. Moment Generating Functions
1. Find the MGF of the Bernoulli(p) and the Binomial(n, p) distributions. Verify that the Binomial is the sum of n i.i.d. Bernoulli(p) RVs using the MGF method.
Solution: Let X1 , . . . , Xn be i.i.d Bernoulli(p
STAB52H3 Week 5 Quiz _ _ _ _
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STAB52 Quiz 1
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1. How many ways are there to permute the letters in the word T EN N ESSEE?
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2. A certain family have 8 children, consisting of 4 boys and 4 girls. Assuming that all birth orders are equally li
STAB52H3 Quiz 1
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1. How many ways are there to permute the word NONNEGATIVE? [3]
2. There are n people in a room. Assume each persons birthday is equally likely to be any
of the 365 days of the year and the peoples birthday ar
STAB52 - Introduction to Probability Quiz 2 (week 5)
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1. Let X Geometric(p), find P (X 2)
STAB52 - Introduction to Probability Quiz 4 (week 9)
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Problem Set 5. Random Variables
1. Suppose that we roll two fair six-sided dice, and let Y be the sum of the two numbers showing.
Compute P (Y = y) for every real number y.
2. Suppose that a bowl contains 100 chips: 30 are labelled 1, 20 are labelled 2, a
Problem Set 6. Discrete Distributions
1. Suppose a symmetrical die is rolled 20 independent times, and each time we record whether or
not the event cfw_2,3,5,6 has occurred.
i What is the distribution of the number of times this event has occurred in the
Problem Set 4. Independence
1. Consider a standard 6-sided die and a biased die with sides (1,2,3,4,6,6), i.e. one which has no
5 and two 6s. Assume you pick one of the two dice, with equal probability, and roll it twice.
Define the events A1 = cfw_1st ro
Problem Set 8. Important Distributions
1. Let X NegBinom(r, p) and Y NegBinom(s, p), where X and Y are defined on separate
and independent Bernoulli(p) trial sequences. If Z = X + Y , what is the distribution of Z?
Justify you answer.
2. (Hypergeometric d
Problem Set 7. Continuous Distributions
1. Let W Uniform(1,4). Compute each of the following:
i P (W 5)
ii P (W 2)
iii P (W 2 9) (Hint: If W 2 9, what must W be?)
iv P (W 2 2)
(
1/3
Solution: The PDF of the Uniform(1,4) is f (x) =
0
i P (W 5) =
R
R
0dx =
Problem Set 5. Random Variables
1. Suppose that we roll two fair six-sided dice, and let Y be the sum of the two numbers showing.
Compute P (Y = y) for every real number y.
Solution: Define sample space S = cfw_(1, 1), (1, 2), (1, 3), . . . (6, 6) contain
Problem Set 3. Conditional Probability
Instructor: Prof. Sotirios Damouras
Teaching Assistant: Mohsen Soltanifar
September 17, 2016
1. Suppose we deal five cards from an ordinary 52-card deck.
i What is the conditional probability that all five cards are
Problem Set 8. Important Distributions
1. Let X NegBinom(r, p) and Y NegBinom(s, p), where X and Y are defined on separate
and independent Bernoulli(p) trial sequences. If Z = X + Y , what is the distribution of Z?
Justify you answer.
Solution: The NegBin
Problem Set 12. 2D Integrals
1.
i. Set up a double integral of f (x, y) over the part of the unit square, 0 x 1, 0 y 1,
on which y x/2.
ii. Set up a double integral of f (x, y) over the part of the unit square on which x + y > 1/2.
iii. Set up a double in