Question

Andy is always looking for ways to make money fast. Lately, he has been trying to make money by gambling. Here is the game he is considering playing: The game costs 2 dollars to play. He draws a card from a deck. If he gets a number card (2-10), he wins nothing. For any face card (jack, queen or king), he wins 3 dollars. For any ace, he wins 5 dollars and he wins an extra $20 if he draws the ace of clubs.

a) if x = money gained, write down the pmf for this game.

b) What are the E(X) and V(X)?

C) Find and sketch the cdf, F(x)

Answer 1

a)

total cards=52

outcome	X	P(X)
number card=36	0-2 = -2	36/52 = 0.6923
face card=12	3-2 = 1	12/36 = 0.2308
ace except ace of club=3	5-2 = 3	3/52 = 0.0577
ace of club=1	25-2 = 23	1/52 = 0.0192

b)

X	P(X)	X*P(X)	X² * P(X)
-2	0.6923	-1.38	2.77
1	0.2308	0.23	0.23
3	0.0577	0.17	0.52
23	0.0192	0.44	10.17

	P(X)	X*P(X)	X² * P(X)
total sum =	1	-0.53846	13.69

mean = E[X] = Σx*P(X) =            - $0.54

E [ X² ] = ΣX² * P(X) =            13.6923
          
variance = E[ X² ] - (E[ X ])² =            13.4024

c)

X	P(X)	F(x)
-2	0.6923	0.6923
1	0.2308	0.9231
3	0.0577	0.9808
23	0.0192	1.0000

please revert for doubts