Question

5. (15 pts) a) A coin is tossed 5 times. Let X be the number of Heads on the first 4 tosses and Y be the number of Heads on the last three tossed. Find the joint probabilities Pij = P(X = 1, Y = j) for all relevant i and j. Find the marginal probabilities pit and p+, for all relevant i and j. b) Find the value of A that would make the function Af(x,y) a PDF. Where 3+x2 - yy, for 1<x<3 and 0 <y <1 f(x, y) = 0 otherwise

Answer 1

(a) A fair coin is tossed 5 times. No- of all possible We have to find A 9 A-fray will be a bit 0,0.0. 5.) (6) fresy) - 1 3 + x ² - 4 1 5X +39 0 EYLt Then we must have 1 dx = 1 J J A frany de dy = 1 m (front-pa) de [sx + 2y - you I do 4 1 (22 +224) dx = 1 ) AT & AL? +4414 = A 1 AA Э) А - A. A = 85 - A = 6085 sty Cases = 2 = 32, each of them is equally likely sample VWE2= space. E-l. Peobability P[{w}] = 322 X: Number of heads on 1st 4 torses. last 3 tosses Clearly x takes values = 0,1,2,3,46 Y takes values 50,1,2,3, individually. y heads on U

heb, ks = [ X2,44] =P[4=d, Y=3]. #41,5). Paj = P[4=;] = I pass Marginal probability of y. ki P[x=2, 4 =]] – Joint probability of (X,Y). ht= P[x=XiJ = P[w: il = [ Pis - Marginal distri- Now Poo = P[x = 0, y = 0] = Pes[ No heads in 1st 4 toss and last 3 tosses 1 2 PLL HTTTHEJ+P [STHT TH] PoL =P[*0. Y=4] - P[277] : P\*=0,4-2] -0, tos-f(x="X-3] =-0. 680 = P(x-4, Y=o] =P[HTTTT}}+THT+7) 21 = P[x=1,4-43 =P +PILTTHTT}]+P[ RTTTATES La = P(x=+*+2) = PBTTTHH]]<P[TT HT44] 13 =P [x = 2, Y = 3] = 0, $20= P(x=2,1=0] b-fx-2,71] = [WATTH}]+PEST +P[2417H T}]+![4TH HTT}]+[TATATE 4. 32 2 32 32

Joint probability dostea butim of (x,y)! – 4 ) PC y =j] = 6+j 6 182. S 2 3 S () 12ม ส่วน 13 1/3 ม -) 52ม ปรม 6) 113 0 ม่วง หวง 42ม 3432 4142ม 2 D () 375 52. 4(22) 4|2- P[4] 52 *22, 22 32. 2452 1 - Pit