Question

help with number 4

4. Roll a die and flip a coin. Let Y be the value of the die. Let Z = 1 if the coin shows a head, and Z = 0 otherwise. Let X = Y + Z. Find the variance of X. 5. (a) If X is a Poisson random variable with = 3, find E(5*).

Answer 1

Question no Rolla la die die an of the 2=0 die set otherwise and flir a coin 2=1 if the coin het x-x+7 . Let y be the value shows head, and find the varion e of varion e of x. . 2=1 net x= Otherwise - When a Rollia die then Roll in die then its sample space be S- $1,2,3,4,5,6 , and for fliping a coins its sample space be Sz= {HIT) Here the Joint nondo variable x- (Y,Z ). hose outcomes are ordered pairs of the form (4,2) more oL2L1, and Icych Here Y and Z are independent so X is. briform with I assigned to each outcome. Now E(X)=E(y)+ E(3), since X=Y+Z so E(X)=my - Zusfi , mere tie is the probability mass for at y = yi, fi- %, 1=1,2, s. So E(X) - È xi ft) - t vist (1+2+3+4+5+6) - 2 - Now ECZ) = 3 zis sit (probability of getting Head Probability of getting tail) so E(2) 01 + 0 1 – 0 atptai 2. E(X) = E(Y) + E(2): += 4 - 3 Mean of x= x= E(X)= 4 . variance of t vary + varz,. obo vary= (yi- mysti cimiti - (35-4/2)45 2 (1-7/2 + (2-7/2) + (3-772) + (4-7/22 +(5-2) 4 (6-7/22] 6" x 70 = 35 - © + 9+1+1+ 9 +25]

- Now variance of zo Š ci-malt = {(1-12 + 1 2 + (0-)=1} Tors from (2) = g + $} = - Finally from 1 and ② we get vary= 35 and varza | so the required variance of x is var x= varyt var z 35+3 2 2573 - 7 - 12 The required variance of
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