Question

Bill pays $1 for the privilege of rolling a fair die. If an odd number shows up, Bill will win as many dollars as the number that shows up on the die. If an even number shows up, he will lose $3.

Find the following:

a) The probability function of Bill’s net winnings.

b) Bill’s expected net winnings and the standard deviation of these winnings.

Answer 1

a)

below is probability mass function of Bill's net winning"

for odd numbers ; net winning =odd number-1

P(X=0)=1/6

P(X=2)=1/6

P(X=4)=1/6

P(X=-4)=P(even number)=3/6=1/2

b)

x	P(x)	xP(x)	x2P(x)
0	1/6	0.000	0.000
2	1/6	0.333	0.667
4	1/6	0.667	2.667
-3	1/2	-1.500	4.500
	total	-0.500	7.833
	E(x) =μ=	ΣxP(x) =	-0.5000
	E(x2) =	Σx2P(x) =	7.8333
	Var(x)=σ2 =	E(x2)-(E(x))2=	7.583
	std deviation=	σ= √σ2 =	2.7538

from above expected winning =-0.5

standard deviation =2.7538