IEOR E4404 Assignment 1 Simulation, Spring 2015

Assignment 1

Problem 1.

Suppose that there are 20 different types of coupons and you wish to collect all of them.You collect one coupon every day, and it is equally likely for you to collect any of the 20 types.

(a) What is the expected number of distinct coupon types that you obtain in 60 days?

(b) Let X be the number of coupons that you collect until you have all 20 types. What is the expectedvalue of X? What is the variance of X?

Problem 2.

At a certain stage of a criminal investigation, the inspector in charge is 60% convincedof the guilt of a certain suspect. Suppose now that a new piece of evidence that there is a new piece ofevidence that shows the criminal is left-handed. If 20% of the population is left-handed, how certain ofthe guilt of the suspect should the inspector now be, if it turns out that the suspect is also left-handed?

State clearly any assumptions that you make.

Problem 3.

(a) Show that n0+n1+ · · · +nn= 2n,where nk= n!/[(n ? k)!k!].(b) X be a negative binomial random variable with parameters (r, p) and Y a binomial r.v. withparameters (n, p).

Show thatP(X > n) = P(Y < r).

Problem 4.

Random variables X and Y are said to have a bivariate normal distribution with parameters(µ1, µ2, ?21, ?22, ?) if their joint density function f(x, y) is given byf(x, y) = 12??1?2p1 ? ?2e? 12x?µ1?12?2?x?µ1?1·y?µ2?2+y?µ2?22/(1??2), x, y ? R.

What is the conditional probability density function of X given Y = y? What about that of Y givenX = x?

Problem 5.

For Unif(0, 1) random variables U1, U2, . . . defineN1 = Minimum (n :Xni=1Ui > 1).That is, N1 is equal to the number of random numbers that must be summed to exceed

1. EstimateE[N1] by generating 100, 1000 and 10,000 values of N1 respectively. What do you think is the value ofE[N1]?Do the same for N2, the number of random numbers that must be summed to exceed

2. That is,N2 = Minimum (n :Xni=1Ui > 2),and estimate E[N2] by generating 100, 1000, and 10,000 values of N2 respectively.1-1

Problem 6.

(Exercise 10 in Chapter 4, “Simulation” by Ross, 4th edition) In this problem we willgenerate a negative binomial random variable with parameters (r, p) in three different ways. (You donot need to implement your codes, but you need to present your precise algorithms and provide clearreasoning.)Recall that a NB(r, p) r.v. has p.m.f given byp(k) = k ? 1r ? 1pr(1 ? p)k?r, k = r, r + 1, . . .

(a) Use the relationship between NB(r, p) and Geom(p), and the relationship between Geom(p) andUnif(0, 1) taught in class (or equivalently, in Example 4d, Chapter 4 of the textbook) to obtainan algorithm to generate NB(r, p).

(b) Verify the relationp(k + 1) = k(1 ? p)k + 1 ? rpk.

(c) Use the relation in part (b) to give a second algorithm for generating NB(r, p).

(d) Using the interpretation that NB(r, p) counts the number of i.i.d Bern(p) trials required to accumulater successes, obtain yet another approach for generating NB(r, p).

Problem 7.

(Bonus) Verify the conditional variance formula. Namely, for any two random variablesX and Y , show thatV ar(X) = E[V ar(X | Y )] + V ar(E[X | Y ]).1-2