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1. Let Y be binomial(15...

1. Let Y be binomial(15, 1/3). Evaluate V ar(Y ). Note: You must show

your work.

2. Let X be a uniform((0, 1)) rv. Set Y = −log(X). Calculate the pdf

of Y .

3. Three people each roll a fair 6 sided die. Calculate P(at least two of the

dice show the same number). Repeat the same process with 4 people.

Is this similar to the birthday problem?

4. Show E(|X|) = 0 implies P(X = 0) = 1.

5. Let A and B be independent events. Show that A and Bc are also

independent.

6. Let P(A) = P(B) = 1. Show P(AB) = 1.

7. Let X and Y be independent Poisson rv’s. Show that X+Y is Poisson.

8. Let X ⇠ Poisson(2) be independent of Y ⇠ Poisson(4). Set W =

X + Y . Calculate P(X = k|W = 3) for k = 0, 1, 2, 3.

9. Let X ⇠ binomial(10, p) be independent of Y . If X+Y ⇠ binomial(15, p)

show Y ⇠ binomial(5, p).

10. Let Z1, Z2, . . . be iid Bernoulli(1/3) and let Sn = Z1 + · · · + Zn. Let

T denote the smallest n such that Sn = 2. Calculate V ar(T).

11. Toss a fair coin. If H obtains you select a chip from Hat#1. Otherwise

you select 3 chips without replacement from Hat#2. Hat#1 contains

3 red chips and 4 black chips while Hat#2 contains 5 reds and 2 black

chips. Let A={at least 1 red chip is selected}. Calculate P(H|A).

12. Let X ⇠ geometric(1/3). Calculate P(X >1) and V ar(X).

13. A rv X has pgf given by G(s) = E(sX) = .1s + .4s4 + .5s16. Calculate

E(pX).

14. Two fair 6-sided dice are rolled. Let X and Y denote the number of

dots showing on each die. Let M = max{X, Y }. Calculate P(M <=2).

15. Suppose An " A or An # A. Show P(An) ! P(A). Note: You must

first find A.

your work.

2. Let X be a uniform((0, 1)) rv. Set Y = −log(X). Calculate the pdf

of Y .

3. Three people each roll a fair 6 sided die. Calculate P(at least two of the

dice show the same number). Repeat the same process with 4 people.

Is this similar to the birthday problem?

4. Show E(|X|) = 0 implies P(X = 0) = 1.

5. Let A and B be independent events. Show that A and Bc are also

independent.

6. Let P(A) = P(B) = 1. Show P(AB) = 1.

7. Let X and Y be independent Poisson rv’s. Show that X+Y is Poisson.

8. Let X ⇠ Poisson(2) be independent of Y ⇠ Poisson(4). Set W =

X + Y . Calculate P(X = k|W = 3) for k = 0, 1, 2, 3.

9. Let X ⇠ binomial(10, p) be independent of Y . If X+Y ⇠ binomial(15, p)

show Y ⇠ binomial(5, p).

10. Let Z1, Z2, . . . be iid Bernoulli(1/3) and let Sn = Z1 + · · · + Zn. Let

T denote the smallest n such that Sn = 2. Calculate V ar(T).

11. Toss a fair coin. If H obtains you select a chip from Hat#1. Otherwise

you select 3 chips without replacement from Hat#2. Hat#1 contains

3 red chips and 4 black chips while Hat#2 contains 5 reds and 2 black

chips. Let A={at least 1 red chip is selected}. Calculate P(H|A).

12. Let X ⇠ geometric(1/3). Calculate P(X >1) and V ar(X).

13. A rv X has pgf given by G(s) = E(sX) = .1s + .4s4 + .5s16. Calculate

E(pX).

14. Two fair 6-sided dice are rolled. Let X and Y denote the number of

dots showing on each die. Let M = max{X, Y }. Calculate P(M <=2).

15. Suppose An " A or An # A. Show P(An) ! P(A). Note: You must

first find A.

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