Question 2) Two fair ten-sided dice are tossed independently. Let M be the RV as

Question

Question 2)

Two fair ten-sided dice are tossed independently. Let M be the RV associated with the greatest number of the two tosses (for example, if the outcome of the first toss is 7 and the outcome of the second is 10, then M = 10).

a) Compute Pr [M = 1].

b) Find the pmf of M.

c) Compute E[M].

-------> I pay you 3 dollars if M 5. You pay me 1 dollar otherwise (if M > 5).

d) Find the pmf of Y , where Y is the RV associated with the outcome you get playing this game.

e) Is this game fair? Find E[Y].

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Can you help me with c, d, e please? Thank you!

Explanation / Answer

2) here let X is outcome on first die and Y on second die.

here probabilty distribution of M is given as below

Expected M =E(M) =7.15

d)as if M<=5 ; Y=3 ; hence P(Y=3) =P(M=1)+P(M=2)+P(M=3)+ P(M=4)+P(M=5)=0.25

and for M>5 ,Y=-1 ; P(Y=-1) =P(M=1)+P(M=2)+P(M=3)+ P(M=4)+P(M=5)=0.75

e) expected value of game =3*P(Y=3)+(-1)*P(Y=-1) =3*0.25-1*0.75 =0

as expected value of game is 0, the game is fair.

M P(M) M*P(M) 1 0.01 0.01 2 0.03 0.06 3 0.05 0.15 4 0.07 0.28 5 0.09 0.45 6 0.11 0.66 7 0.13 0.91 8 0.15 1.2 9 0.17 1.53 10 0.19 1.9 Total 7.15

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