you roll two 4 sided fair die having sides numbers from 1-4 Panther ID: Fall 201

Question

you roll two 4 sided fair die having sides numbers from 1-4 Panther ID: Fall 2017 STA 2023-Statistics for Economics and Business Quiz #2 There are 2 problems for a total of 20 points. Show your work. Please write your work neatly Problem 1 (10 points). You roll two 4-sided (pyramid-shaped) fair die, having sides numbered fra Define the following random variables X and Y x = (The smallest value of the two die) Y (The sum of the two die faces a) Compute and graph the PDF of X and compute E(X). b) Suppose you will win $1 if Y> 5 and lose $1 if Y s 5. Would you place this bet?

Explanation / Answer

a)

Total number of possible outcomes = 4*4 = 16

PDF of X:

P(X=1) = 7/16...... this includes outcomes - (1,1) (1,2) (1,3) (1,4) (2,1) (3,1) (4,1)

P(X=2) = 5/16.....  this includes outcomes - (2,2) (2,3) (2,4) (3,2) (4,2)

P(X=3) =3/16 ......this includes - (3,3) (3,4) (4,3)

P(X=4) = 1/16....i.e. (4,4)

E(x) = 1*(7/16) + 2*(5/16) + 3*(3/16) + 4*(1/16) = 30/16 = 15/8

b).

PDF of Y:

P(Y=2) = 1/16...... this includes outcome - (1,1)

P(Y=3) = 2/16.....  this includes outcomes - (1,2) (2,1)

P(Y=4) =3/16 ......this includes - (1,3) (3,1) (2,2)

P(Y=5) = 4/16....this includes - (1,4) (2,3) (3,2) (4,1)

P(Y=6) =3/16....this includes - (2,4) (3,3) (4,2)

P(Y=7) = 2/16.....(3,4) (4,3)

P(Y=8) = 1/16.....(4,4)

P(Y > 5) =3/16 + 2/16 + 1/16 = 6/16

P(Y <= 5) = 1-6/16 = 10/16

Since probability of loosing $1 is more than that of winning $1, one would not place the bet.

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