ISyE 3044 Fall 2017 Test #1a Solutions

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1 1 NAME ISyE 344 Fall 217 Test #1a Solutions This test is 75 minutes. You re allowed one cheat sheet. Good luck! 1. Suppose X has p.d.f. f(x) = 4x 3, < x < 1. Find E[ 2 X 2 3]. Solution: By LOTUS, we have [ 1 ] E X 2 = 1 x 2 4x3 dx = 4x dx = 2. This implies that [ 2 ] E X 3 2 [ 1 ] = 2E 3 = 1. X 2 2. Suppose that X has p.d.f. f(x) = 2x, x 1. What s the p.d.f. of the random variable e X? Solution: Let Y = e X. The c.d.f. of Y is G(y) = P(Y y) = P(e X y) = P(X ln(y)) = ln(y) 2x dx = [ln(y)] 2. This implies that the p.d.f. of Y is g(y) = d 2ln(y) G(y) =, 1 y e. dy y 3. Toss two dice and observe their sum. What is the expected number of tosses until you observe a sum of 7? Solution: X Geom(1/6), so E[X] = 6.

2 2 4. Consider a Poisson process with rate λ = 2. What is the distribution of the time between the 5th and 7th arrivals? Solution: The times between Poisson arrivals are i.i.d. Exp(2). Thus, the sum of the 2 interarrival times between the 5th and 7th arrivals is Erlang 2 (2). 5. YES or NO? If X and Y are independent normal random variables, are they necessarily uncorrelated? Solution: Yes. 6. Fill in the blank. A Poisson process has i.i.d. interarrivals. Solution: exponential. 7. Suppose that X and Y have joint p.d.f. f(x, y) = 8xy for < x < y < 1. Find E[X]. Solution: We have f X (x) = f(x, y) dy = R x so that E[X] = xf X (x) dx = R 8xy dy = 4(x x 3 ), < x < 1, 4(x 2 x 4 ) dx = 8/ Let X and Y be i.i.d. Exponential with rate 2. Find P(X + Y 1). Solution: X + Y Erlang 2 (2). So P(X + Y 1) = k 1 i= e λx (λx) i i! = 1 e 2(1) (2(1)) i i= i! = 3e If X and Y are i.i.d. Nor(,1) random variables, find P(Y + 1 < 2X).

3 3 Solution: Note that E[Y 2X] = and Var(Y 2X) = Var(Y ) + 4Var(X) = 4, so that Y 2X Nor(, 5). Thus, P(Y + 1 < 2X) = P(Y 2X < 1) = P(Nor(, 5) < 1) = P(Nor(, 1) < 1/ 5) = Φ( 1/ 5) = Φ(.447) = If X and Y are both Normal(4,1) with Cov(X, Y ) = 6, find Var(X Y ). Solution: Var(X Y ) = Var(X) + Var(Y ) 2Cov(X, Y ) = BONUS: What does Dave really enjoy? (a) An intellectual discussion with a typical UGA student. (b) A lovely cup of piping hot coffee to start the day. (c) Attending a Justin Bieber concert. (d) Attending a Justin Bieber concert at UGA. Solution: (b). 12. Suppose that the Atlanta Hawks play i.i.d. games, each of which has win probability.7. Let X be the number of games until the Hawks achieve their first win. Find the smallest x such that P(X x).95. Solution: The number of games until the first win is X Geom(.7), so that P(X = x) = q x 1 p = (.3) x 1 (.7). It can also be shown that the c.d.f. is F (x) = P(X x) = 1 q x = 1 (.3) x (though you don t need to know this if you do a trial-and-error argument to find the smallest x). Now, F (x) = 1 (.3) x.95 iff.5 (.3) x, which is achieved by x = 3.

4 4 13. Suppose that X 1,..., X 1 are i.i.d. with values 1 and 1, each with probability.5. (This is a simple random walk.) Find the approximate probability that the sum 1 X i will be at least 1. Solution: Note that E[X i ] = and Var(X i ) = E[Xi 2 ] (E[X i ]) 2 = 1. Then the Central Limit Theorem implies that 1 X i Nor(, 1), and so X i 1) P 1 P( ( Z 1 ) 1 = P(Z > 1) = Consider the linear congruential pseudo-random number generator X i+1 = (3X i + 1)mod(8). Using X = 1, calculate the fifth pseudo-random number U 5. Solution: We have X 1 = 4, X 2 = 5, X 3 =, X 4 = 1, and so X 5 = 4. Thus, U 5 = 4/8 = If U Unif(, 1), what s the distribution of 3ln(U 2 )? Solution: Note that 3ln(U 2 ) = 6ln(U) Exp(1/6). 16. If U 1, U 2, and U 3 are i.i.d. Unif(,1), what s the distribution of 3ln(U 1 U 2 (1 U 3 ))? Solution: Note that 3ln(U 1 U 2 (1 U 3 )) 3ln(U 1 U 2 U 3 ) Erlang 3 (1/3) (or gamma). 17. If U 1 and U 2 are i.i.d. Unif(,1), what s the distribution of 2 2ln(U 1 )cos(2πu 2 )? Solution: By a homework problem, this is Nor(, 4). 18. Suppose that you want to estimate the integral I = [1 cos(πx)]dx.

5 5 The following numbers are a Unif(,1) sample: Use the Monte Carlo method from class to approximate the integral via the estimator Ī4. Solution: Ī 4 = b a n = 1 4 = n g(a + (b a)u i ) 4 g(u i ) = [1 cos(πx)] 19. Consider the differential equation f (x) = (x 3)f(x) with f() = 1. Use Euler s method with increment h =.1 to find the approximate value of f(.2). Solution. You can actually get the true answer using separation of variables, and it turns out to be f(x) = exp{ x2 2 3x}. But our job is to use Euler to come up with an iterative approximation, so here goes. As usual, we start with f(x + h) = f(x) + hf (x) = f(x) + h(x 3)f(x) = f(x)[1 + h(x 3)], from which we obtain the following table. x Euler approx true f(x) Thus, the desired Euler approximation for f(.2) is.941.

6 6 2. Joey works at a chocolate store. Starting at time, we have the following 4 customer interarrival times (in minutes): Customers are served in LIFO fashion. The 4 customers order the following numbers of chocolate products, respectively: Suppose it takes Joey 3 minutes to prepare each chocolate product. Further suppose that he charges $3/chocolate. Unfortunately, the customers are unruly and each customer causes $.5 in damage for every minute the customer has to wait in line. (a) What is the average number of customers in the system during the first 1 minutes? (b) How much money will Joey make or lose with the above 4 customers? Solution: Consider the following table. cust intrarrl arrl time serv start serve time depart wait sys time (a) Let X i denote the amount of time Customer i spends in the system during the time interval [,1]. In particular, X 1 = 1, X 2 = 1 2 = 8, X 3 = 3, and X 4 = 1. The average number of customers in the system during the first 1 minutes is total customer time 1 = = 22 1 = 2.2. (b) Joey makes 3( ).5( ) = $1.5.