Strategic Practice and Homework 8 - Projects at Harvard

Stat 110 Homework 8, Fall 2011Prof. Joe Blitzstein (Department of Statistics, Harvard University)1. Let X be the number of distinct birthdays in a group of 110 people (i.e., thenumber of days in a year such that at least one person in the group has that birthday).Under the usual assumptions (no Feb 29, all the other 365 days of the year are equallylikely, and the day when one person is born is independent of the days when the otherpeople are born), find the mean and variance of X.Let I j be the indicator r.v. for the event that at least one of the people was born on thejth day of the year, so X = P 365j=1 I j with I j ⇠ Bern(p), where p =1 (364/365) 110 .The I j ’s are dependent but by linearity, we still haveE(X) =365p ⇡ 95.083.By symmetry, the variance is✓ ◆ 365Var(X) =365Var(I 1 )+2 Cov(I 1 ,I 2 ).2To get the covariance, note that Cov(I 1 ,I 2 )=E(I 1 I 2 ) E(I 1 )E(I 2 )=E(I 1 I 2 ) p 2 ,and E(I 1 I 2 )=P (I 1 I 2 =1)=P (A 1 \ A 2 ), where A j is the event that at least oneperson was born on the jth day of the year. The probability of the complement is✓ ◆ 110 ✓ ◆ 110 364 363P (A c 1 [ A c 2)=P (A c 1)+P (A c 2) P (A c 1 \ A c 2)=2,365 365so Var(X) =365p(1 p)+365· 364 · (1 (2 364365110 363365110) p 2 ) ⇡ 10.019.2. A scientist makes two measurements, considered to be independent standardNormal r.v.s. Find the correlation between the larger and smaller of the values.Hint: note that max(x, y)+min(x, y) =x + y and max(x, y) min(x, y) =|x y|.Let X and Y be i.i.d N (0, 1) and M =max(X, Y ),L=min(X, Y ). By the hint,E(M)+E(L) =E(M + L) =E(X + Y )=E(X)+E(Y )=0,E(M) E(L) =E(M L) =E|X Y | = 2 p ⇡,where the last equality was shown in class. So E(M) =1/ p ⇡,andCov(M,L) =E(ML) E(M)E(L) =E(XY )+(EM) 2 =(EM) 2 = 1 ⇡ ,1