Question

6. Let X and Y denote the number of times Harry Potter and Ron Weasley manage to irritate Professor Snape in one day, respectively. Suppose that X and Y have the following joint probability distribution: x у 0 2 در 4 0 0.15 0.10 0.05 0 0 0.10 0.10 0.20 0.10 0 2 0 0 0.05 0.10 0.05 Px(x) 0.25 0.20 0.30 0.20 0.05 Recall Homework 4: E(X)= 1.6, SD(X)= 1.2. f) Find the probability that Harry Potter irritates Professor Snape more times in one day than Ron Weasley. That is, find P(X>Y). Find Py(y), the marginal p.m.f. for the number of times Ron Weasley manages to irritate Professor Snape in one day. h) Each day, Professor Snape takes 10 points from Gryffindor, simply because he can. Additionally, Professor Snape takes 15 points from Gryffindor each time Harry Potter irritates him, and 5 points each time Ron Weasley irritates him. If these are the only three sources of point deductions for Gryffindor, what is the expected point loss for Gryffindor each day? i) Are X and Y independent? If not, find p = Corr(X, Y).

Answer 1

f> P(x >7) 91 Y = 0 pltxo) = P(x-1) + PLx - 2) + + P(x =4) 0.1 + 0.05 0.15 - If Yo1 Px2)) = P(x=2) + Pl x= 3) + P(x-4) 0.2 + 0.1 = 0.3 9+ y = 2 P ( x 720 - P ( X =3) + P ( X =4) 0:1 + 0.05 0.15 so, P (X>7) 0.6 u 0:15 + 0.3 + 0:15 g) byly) is given by y Þy (y = o) - Ž $(x,0) x=0 O. 15 + 0.10 + 0 05 0.3 = 4 þy (y=2 - E bcn,i) x = 0 = 0.10 + 0:10 + 0:20 + 0,10 0.5 4 by ly=2) blu, 2) x0 0:05 + 0.10+ 0.05 0.2

h2 E(X) - 1.6 ECY) O x 0.3 + 1 X 0.5 + 0.2 2 x = - 0.9 Expected point loss for gryffindor each day is 15 X 1.6 + 5 X 0.9 + 10 38.5 The expected value of is XXY ECXY) Eb (ny). W.Y - 0:10 + 0.40 + 0:30 + O. 20 + 0.60 + 0.40 or Y=0 . 2 ( all terms with. either x=0 pcm, y = 0 will result in on not been shown) hence has ECX ) - 1.6 ECY) 0.9 ЕСхТЕСТ 1.6 x o.9 1.44 + ECXY] = Hence, xay are not independent. con(x,y) 3 ECxy] ECXIE CY) - 2 1:44 0:56 SD CX3 - 1.2 we need to find the standard eviation of Y Ely? ] 0²x 0.3 + 12 xo 5 + 2? * 0.2 0.5 + 0 8 1.3 Var [y] ECY? I - ECY]² - 1:3 0.8) 0:49 Sncy 0:49 - 0.7 Covlxiy) corr(x,y) - 0.56 e 0.667 o+ oy 1.2 xo.7 Thank You !! Please upvote!!