Question

In the game Monopoly, a player rolls two dice on his or her turn. To make the discussion easier, let’s
assume one die is red and the other is blue. After rolling the dice, the player adds the spots on the red
and blue dice, and moves that many squares. There are 40 squares that form a square along the sides
of the board. Let X denote the spots on the red die and Y the spots on the blue die.

(a) X and Y are iid with a certain named distribution. What is the distribution and specify any
parameters associated with the distribution.
(b) What is the moment-generating-function (mgf) of X.
(c) Use the mgf of X to show that E(X) = 3.5.
(d) Confirm (any way you wish) that Var(Y ) = 35/12.
(e) Plot the probability mass function of X + Y .
(f) Find the expected value of a move; that is, find E(X + Y ).
(g) Find the standard deviation of a move.
(h) What is the chance that the player moves 1 square? 7 squares?
(i) What is the chance of rolling doubles?
(j) Find P(X + Y ≥ 5). What is this asking for in words?
(k) What is the chance of at least one of the dice being a six?

Answer 1

Two dice are rolled.

The sample space is

S = { (i, j) / i =1:6, j =1:6}

a) Let X : spot on the red die. ( first die)

and Y : Spot on the second die.

X takes values 1,2,3,4,5,6 each with probability 1/6

i.e. X follows discrete uniform distribution.

X ~ Discrte uniform ( n = 6)

The p.m.f. of X is

P ( X =x) = 1/6 , x= 1:6

Similarly Y ~ Discrete Uniform ( n = 6)

P ( Y=y) = 1/6 ; y =1:6

b) Moment -generating Function of X

c) To find Expected value of X by using m.g.f.

d) Var ( Y ) = E ( Y²) - (E(Y))²

Since

E (Y) = 21 / 6

Aliter :

Since Y ~ Discrete Uniform ( n=6)

Var (Y) = n²-1 /12 = 6²-1 /12 = 35/12

e) Let Z =X + Y

i.e Sum of spots on two dice.

Z takes values 2, 3, ,4, ........., 12

The probability distibution of Z is

Z	2	3	4	5	6	7	8	9	10	11	12	Total
P(Z=z)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36	1

by using R

> z= 2:12
> p = c(1/36,2/36,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36)
> plot(z,p,xlab="Z", ylab="Probability",main="P.M.F. of X +Y", "l")

f) Expected value of move

E ( X + Y ) = E ( Z ) = sum ( Z * P(Z))

= 7

g) Var ( X + Y ) = Var ( Z )

Var (Z) = E(Z²) - ( E(Z))²

E(Z²) = sum ( Z² * P(Z))

E(Z²) = 54.8333

Var(Z) = 54.8333 -49

= 5.8333

SD(Z) = sqrt(5.8333) = 2.4152.

Standard deviation of move is 2.4152.