Question

Let X be a discrete random variable with P(X = 1) = 1 4 1 P(X = 2) = 8 1 P(X = 3) = 2 P(X = 4) = 8 Then given X = x, we roll a fair 4-sided die x times. (The 4- sided die is equally likely to come up a 1, 2, 3, or 4). Let y be the number of times we roll a 1. (a) Find E[Y| X]. Hint: Remember E|Y|X] is a random variable, so X will be part of the answer. (Or if you prefer, you can list out the different cases.) (b) Find E[Y] Hint: We didn't have any homework from this section, so for similar problems you can look at Example 2 from the class notes for 10.3, or look at #6 on review 3.

Answer 1

we can find By the free diagram PLY= y). Y = y X = X 3/4 Yy Yy Y8 0 V16 3/8 2 1 2 Ya 27/64 3 27/64 9/64 2 164 8/256 o 27/64 54/256 3164 2 1/296 3 27 + -in 27 + I 81 8 256 1041 2048 8 16 + + 1. 3 8 II I. 27 2 64 27 64 191 512 4 8 on By Total Probablity Law; P (Y=0)= 1.3 3+1.9 4.4 P(Y= 1) = 1 4 + 1 P (Y=2) = 1 1 1 + P (Y=3)=1. I g 64 P (Y=4) = + 107 1024 + 2 64 L.sy 256 7 512 + 1 los 3 64 - ㅗ 2048 . 8 256

as. By the formula: E(Y(x=x) = { y p (Y=y|x=4) Woon - [y. Ey Plyxyxa P(X=) y y - P (Yay, X=x) P(X=2) By Putting the value of x, y, we get. El 4(x-1)-480 +16) - 5 E (Y | X= 2) = 8 (ot (y (X=3)= = 2.(ot 27 - 3 64 + 2 128 2 + - + 128 3. 128 ) 128 9 E(Y|X=4) = 8. (ot 2 1/2 t 108 2048 + 9 512 4 + 2048 Hence I n=1 2 (Yx*) -ام M=2 3 1=3 n=4

b), E (0) Sฯ ? /-y) 3- | 9| - 6 + 1 + 2 | 63 + 3. 3 +y. (029 512 2018 2 | น 191 S2 + { ราว + 1 6 ฯ 512 LoV8 5 1) 8

Given we roll x is a discrete random variable which denote the no. of times a 4-sided die And its probablity is given by 는 Uz1 1=2 P(X= x) = 8 Nt u=3 ㅗ 8 nay Let y be a random variable which denote the number of times we roll a 1. So, possible outcomes of y are 0, 1, 2, 3, 4. 1, 2, 3, 4 The 4-sided die is fair so getting are equally likely i. e, 7) - P (not getting 1) = 24 P ( getting 3 4