18.440 Midterm 2 Solutions

18.440 Midterm 2: 50 minutes, 100 points 1. (10 points) Suppose that a fair die is rolled 18000 times. Each roll turns up a uniformly random member of the set {1, 2, 3, 4, 5, 6} and the rolls are independent of each other. Let X be the total number of times the die comes up 1. (a) Compute Var(X). ANSWER: npq = 18000(5/6)(1/6) = 2500 (b) Use a normal random variable approximation to estimate the probability that X < 2900. You may use the function 2 a Φ(a) = √12π −∞ e−x /2 dx in your answer. ANSWER: Standard √ derivation is 2500 = 50. Probability X more than 2 standard deviations below mean is approximately Φ(−2). 2. (20 points) Let X1 , X2 , and X3 be independent uniform random variables on [0, 1]. Write Y = X1 + X2 and Z = X2 + X3 . (a) Compute E[X1 X2 X3 ]. ANSWER: Independence implies E[X1 X2 X3 ] = E[X1 ]E[X2 ]E[X3 ] = (1/2)3 = 1/8. 1 (b) Compute Var(X1 ). ANSWER: E(X12 ) = 0 x2 dx = 1/3, so Var(X1 ) = E(X12 ) − E(X1 )2 = 1/3 − 1/4 = 1/12. (c) Compute the covariance Cov(Y, Z) and the correlation coefficient ρ(Y, Z). ANSWER: By bilinearity of covariance, Cov(Y, Z) = Cov(X1 + X2 , X2 + X3 ) = Cov(X1 , X2 ) + Cov(X1 , X3 ) + Cov(X2 , X2 ) + Cov(X2 , X3 ). All terms are zero by independence except Cov(X2 , X2 ) = Var(X2 ) = 1/12. Then ρ(Y, Z) = √ 1/12 (2/12)(2/12) = 1/2. (d) Compute and draw a graph of the density function fY . ANSWER: ∞ fY (a) = −∞ fX (a − y)fX (y)dy where 1 x ∈ (0, 1) . 0 x ∈ (0, 1) fX (x) = Then fX (a − y)fX (y) = 1 a − y ∈ (0, 1), y ∈ (0, 1) . 0 otherwise 1 Now a − y ∈ (0, 1) is equivalent to −y ∈ (−a, 1 − a) or equivalently y ∈ (a − 1, a). Thus fY (a) is equal to the length of the intersection of the intervals (0, 1) and (a − 1, a). This becomes ⎧ ⎪ a<0 ⎪ ⎪0 ⎪ ⎨a 0≤a<1 fY (a) = ⎪ 2−a 1≤a<2 ⎪ ⎪ ⎪ ⎩0a ≥ 2 fY (a) 1 0 2 1 a 3. (20 points) Suppose that X1 , X2 , . . . , Xn are independent uniform random variables on [0, 1]. (a) Write Y = min{X1 , X2 , . . . , Xn }. Compute the cumulative distribution function FY (a) and the density function fY (a) for a ∈ [0, 1]. ANSWER: By independence, P (min{X1 , X2 , . . . , Xn } > a) = P (X1 > a)P (X2 > a) · · · P (Xn > a) = (1 − a)n So FY (a) = 1 − (1 − a)n , and fY (a) = FY� (a) = n(1 − a)n−1 (b) Compute P (X1 < .3) and P max{X1 , X2 , . . . , Xn } < .3. ANSWER: .3 and .3n . (c) Compute the expectation E[X1 + X2 + . . . + Xn ]. ANSWER: By additivity of expectation, this is nE[X1 ] = n/2. 4. (20 points) Aspiring writer Rachel decides to lock herself in her room to think of screenplay ideas. When Rachel is thinking, the moments at which good new ideas occur to her form a Poisson process with parameter λG = .5/hour. The times when bad new ideas occur to her are a Poisson point process with parameter λB = 1.5 per hour. (a) Let T be the amount of time until Rachel has her first idea (good or bad). Write down the probability density function for T . 2 ANSWER: T is exponential with parameter λ = λG + λB = 2, so fT (x) = 2e−2x . (b) Compute the probability that Rachel has exactly 3 bad ideas total during her first hour of thinking. ANSWER: Number N of bad 3 −1.5 ideas is Poisson with rate 1 · λB = 1.5. So P (N = 3) = (1.5)3!e . (c) Let S be the amount of time elapsed before the fifth good idea occurs. Compute Var(S). ANSWER: Variance of time till one good idea is 1/λ2G . Memoryless property and additivity of variance of independent sums gives Var(S) = 5/λ2G = 20. (d) What is the probability that Rachel has no ideas at all during her first three hours of thinking? ANSWER: Time till first idea is exponential with λ = 2. Probability this time exceeds 3 is e−2·3 = e−6 . 5. (20 points) Suppose that X and Y have a joint density function f given by ( 1/π x2 + y 2 < 1 f (x, y) = . 0 x2 + y 2 ≥ 1 (a) Compute the probability density function fX for X. ANSWER: ( √ 1 ∞ 2 1 − x2 −1 ≤ x ≤ −1 fX (x) = f (x, y)dy = π 0 otherwise −∞ (b) Compute the conditional expectation E[X|Y = .5]. ANSWER: Probability density for X given Y = .5 is uniform on √ √ 2 (− 1 − .5 , 1 − .52 ). So E[X|Y = .5] = 0. (c) Express E[X 3 Y 3 ] as a double integral. (You don’t have to explicitly evaluate the integral.) ANSWER: 1 √ 1−x2 −1 √ − 1−x2 1 3 3 x y dydx. π 6. (10 points) Let X and Y be independent normal random variables, each with mean 1 and variance 9. 3 (a) Let f be the joint probability density function for the pair (X, Y ). Write an explicit formula for f . ANSWER: 1 1 1 − (x−1)2 −(y−1)2 2 2 18 f (x, y) = √ e−(x−1) /18 √ e−(y−1) /18 = e . 18π 3 2π 3 2π (b) Compute E[X 2 ] and E[X 2 Y 2 ]. ANSWER: Var(X) = E[X 2 ] − E[X]2 = E[X 2 ] − 1 = 9, so E[X 2 ] = 10. By independence E[X 2 Y 2 ] = E[X 2 ]E[Y 2 ] = 100. 4