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Probability and statistics

Worksheet 4 — Random variable, expectation, and variance

1. A die is thrown twice. Let X1 and X2 denote the outcomes, and deﬁne random variable X to be the minimum of X1 and X2. Determine the distribution of X.

2. A fair die is rolled repeatedly until a six is seen. What is the expected number of rolls?

3. On any given day, the probability it will be sunny is 0.8, the probability you will have a nice dinner is 0.25, and the probability that you will get to bed early is 0.5. Assume these three events are independent. What is the expected number of days before all three of them happen together?

4. An elevator operates in a building with 10 ﬂoors. One day, n people get into the elevator, and each of them chooses to go to a ﬂoor selected uniformly at random from 1 to 10.

(a) What is the probability that exactly one person gets out at the ith ﬂoor? Give your answer in terms of n. (b) What is the expected number of ﬂoors in which exactly one person gets out? Hint: let Xi be 1 if exactly one person gets out on ﬂoor i, and 0 otherwise. Then use linearity of expectation.

5. You throw m balls into n bins, each independently at random. Let X be the number of balls that end up in bin 1.

(a) Let Xi be the event that the ith ball falls in bin 1. Write X as a function of the Xi. (b) What is the expected value of X?

6. There is a dormitory with n beds for n students. One night the power goes out, and because it is dark, each student gets into a bed chosen uniformly at random. What is the expected number of students who end up in their own bed?

7. In each of the following cases, say whether X and Y are independent.

(a) You randomly permute (1,2,…,n). X is the number in the ﬁrst position and Y is the number in the second position. (b) You randomly pick a sentence out of Hamlet. X is the ﬁrst word in the sentence and Y is the second word. (c) You randomly pick a card from a pack of 52 cards. X is 1 if the card is a nine, and is 0 otherwise. Y is 1 if the card is a heart, and is 0 otherwise. (d) You randomly deal a ten-card hand from a pack of 52 cards. X is 1 if the hand contains a nine, and is 0 otherwise. Y is 1 if all cards in the hand are hearts, and is 0 otherwise.

8. A die has six sides that come up with di↵erent probabilities:

Pr(1) = Pr(2) = Pr(3) = Pr(4) = 1/8, Pr(5) = Pr(6) = 1/4. (a) You roll the die; let Z be the outcome. What is E(Z) and var(Z)?

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DSE 210 Worksheet 4 — Random variable, expectation, and variance Winter 2015

(b) You roll the die 10 times, independently; let X be the sum of all the rolls. What is E(X) and var(X)? (c) You roll the die n times and take the average of all the rolls; call this A. What is E(A)? What is var(A)?

9. Let X1,X2,…,X100 be the outcomes of 100 independent rolls of a fair die. (a) What are E(X1) and var(X1)? (b) Deﬁne the random variable X to be X1 X2. What are E(X) and var(X)? (c) Deﬁne the random variable Y to be X1 2X2 + X3. What is E(Y ) and var(Y )? (d) Deﬁne the random variable Z = X1 X2 + X3 X4 +···+ X99 X100. What are E(Z) and var(Z)? 10. Suppose you throw m balls into n bins, where m n. For the following questions, give answers in terms of m and n.

(a) Let Xi be the number of balls that fall into bin i. What is Pr(Xi = 0)? (b) What is Pr(Xi = 1)? (c) What is E(Xi)? (d) What is var(Xi)? 11. Give an example of random variables X and Y such that var(X + Y )6= var(X) + var(Y ). 12. Suppose a fair coin is tossed repeatedly until the same outcome occurs twice in a row (that is, two heads in a row or two tails in a row). What is the expected number of tosses?

13. In a sequence of coin tosses, a run is a series of consecutive heads or consecutive tails. For instance, the longest run in HTHHHTTHHTHH consists of three heads. We are interested in the following question: when a fair coin is tossed n times, how long a run is the resulting sequence likely to contain? To study this, pick any k between 1 and n, and let Rk denote the number of runs of length exactly k (for instance, a run of length k+1 doesn’t count). In order to ﬁgure out E(Rk), we deﬁne the following random variables: Xi = 1 if a run of length exactly k begins at position i, whereink + 1. (a) What are E(X1) and E(Xnk+1)? (b) What is E(Xi) for 1 <i<nk + 1? (c) What is E(Rk)? (d) What is, roughly, the largest k for which E(Rk)1?