Math questions 12 hrs

MATH 4530 – Probability Autumn 2021 Homework 8

Due date: Oct 18, 2021 at 11:59pm You are encouraged to talk with each other, but the work you submit must be your own. You only need to submit your solutions to Problems 1–5, but you should attempt all other problems as well. There will be a quiz on Oct 22, 2021 covering similar (but not necessarily identical) problems.

Problem 1: (2 points) Let X ∼ Geometric(p). Calculate E

( 1

2X ) .

Problem 2 (p.217, #1): (4 points) A coin which lands heads with probability p is tossed repeatedly. Assuming independence of tosses, find formulae for

a) P (exactly 5 heads appear in the first 9 tosses); b) P (the first head appears on the 7th toss); c) P (the fifth head appears on the 12th toss); d) P (the same number of heads appear in the first 8 tosses as in the next 5 tosses).

[You do not need to simplify your answer.]

Problem 3: (4 points) An urn contains n balls labelled 1, . . . , n. We make n independent draws with replacement from the urn and note the number of the ball drawn. Let Xn be the smallest number observed. Show:

a) P (Xn ≥ k) = ( 1 − k−1

n

)n (k = 1, . . . , n);

b) limn→∞ P (Xn − 1 = k) = p(1 − p)k with p = 1 − 1/e. In other words, the limit distribution of Xn − 1 for n →∞ is the Geometric(1 − 1/e) distribution. [Hint: Use ex = lim

n→∞

( 1 + x

n

)n.] Problem 4 (p.218, #7): (5 points) Suppose A and B take turns tossing a biased coin which lands heads with probability p. Suppose A tosses first. Find:

a) the probability that A tosses the first head; b) the probability that B tosses the first head; c) Suppose now that A tosses once, then B twice, then A once, and so on. Repeat a) and b) for

this scheme. d) Find the value of p (under the second scheme) for which A and B have the same probability of

tossing the first head. e) What is B’s chance (under the second scheme) of tossing the first head for very small values of

p? Evaluate this limit as p → 0.

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Problem 5 (p.219, #12): (5 points) Let W1 and W2 be independent geometric random variables with parameters p1 and p2. Find:

a) P (W1 = W2); b) P (W1 < W2); [Hint: Use that if X ∼ Geometric(p), we have P (X > n) = (1 − p)n for every

n ≥ 1 (Homework 6, Problem 9)]. c) P (W1 > W2);

Additional problems that will not be graded and do not need to be submitted:

Problem 6 (p.218, #3): Suppose you pick people at random and ask them what month of the year they were born. Let X be the number of people you have to question until you find a person who was born in December. What is E(X), approximately?

Problem 7 (p.218, #6): The Geometric(p) distribution is often defined as a distribution on {0, 1, 2, . . .} instead of {1, 2, 3, . . .}. A random variable W has Geometric(p) distribution on {0, 1, 2, . . .} if

P (W = k) = qkp (k = 0, 1, 2, . . .).

a) Show that this is the distribution of the number of failures before the first success in Bernoulli(p) trials;

b) Find P (W > k) (k = 0, 1, . . .); c) Find E(W ); d) Find V ar(W ).

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