Question

Please show work :) Will upvote/rate!

3. Discrete Random Variables You have a biased die, where the probability that a number n appears on the die when it is rolled is defined as a random variable X such that Р(X %3D п) — с:п Here c is a positive real number. Now answer the questions below: (a) Find the value of c (b) What is the expected value of the random variable X? (c) Find how close a number you see on the die will be to the expected value on average (i.e., find the variance of X), as defined in L17. (d) If you roll the die 100 times, what is the expected number of die rolls such that the die roll is equal to the previous one? (e) Assume now that your friend has a fair die, while you still have the biased die discussed in points (a-d). You enter a game with your friend. You start rolling the biased die (let's call the outcome X1), then your friend roll the fair die (let's call the outcome Y1), and you continue to alternate in rolling your own die die. The outcome of the game is the series: Х1, Yi, X2, Y2, .. Ху, Y9, X10, Yio, Х11 A point is assigned every time the value of a die roll is equal to the previous one. Your friend gets a point every time that Y = Xi, while you get a point every time Xi+1 = Y. What is the expected number of points of your friend at the end of the game? and what is your expected number of points?

Answer 1

Probability thata nummber defined positive We have a biased die. The afpears on the die wken it u olled RV P(x-n)- en c is m takes Non G valuen A/2,3, 156 thoufone en -1 c- 1 21 aThe value of c is 21 to fmd E (x) EDXI Now, we ao C.Sn G44)(1241) . 22 4.33 E(X) 4.33 E(x)-E(x) E(x)- S Var CX)= Now C En3 m-l 3 21 E(X) = 21- 13. 7189 225 E(x) Var (x)

d) Nov we have 7rolled the die 100times Let the outeomes arce Let wa detine a nen 7andomvaciable 9マミーS1 it xit1=Xi ji-1,2,-.. We are tofind ECz) Now PRi- = FX x] =1-PLXi+ 1- X W34 36 5 36

E(i) 1.FEZI1]to PZi-o 31 36 L1C 1 -31 36