A game is designed as a player pays $5to play the game. He/she then can roll two

Question

A game is designed as a player pays $5to play the game. He/she then can roll two fair dice: a green one and a red one. Let X and Y be two random variable

X=

0 if the green die is 1,2,3,4

          1 if the green die is 5

         2 if the green die is 6

While Y is the actual number observed on the red die. The player will gain dollar amount equaling to X*Y

a) Find the marginal distribution of X and also that of Y

B) Give the joint probability of X and Y by filling in the probability below

C)Find E(X) and E(Y)

x y  

1

2

3

4

5

6

0

1

2

x y  

1

2

3

4

5

6

0

1

2

Problem 4.25 pts) Random variable X has a probability distribution/(x)-X 0cxcl and 0a, Do (a) Prove Rx) is a valid probability distribution function b) Develop the maximum likelihood function of e (c) Show the maximum likelibood estimator (MLE) of 0 is given by (d) Prove that this MLE is also an unbiased estimator of e (e) If a random sample gives x-0.5,-0.6x-0 45,-0.72,-0 62, estimate And what's probability that o.5

Explanation / Answer

Here X = 0 if die show 1,2,3,4

= 1 if die show 5

= 2 if die show 6

Here Y = 1,2,3,4,5,6 as the numbe shows.

(a) Here fX(x) = 4/6 = 2/3 for X = 0

= 1/6 for X = 1

= 1/6 for X = 2

fY (Y) = 1/6 for Y = 1,2,3,4,5,6

(b) Joint probability table

Here f(x,y) = 2/3 * 1/6 = 1/9 for x = 0 and y = 1,2,3,4,5,6

= 1/6 * 1/6 = 1/36; for x = 1,2 and y = 1,2,3,4,5,6

(c) E(X) = 0 * 2/3 + 1 * 1/6 + 2 * 1/6 = 0.5

E(Y) = 1/6 * (1 + 2 + 3 + 4 + 5 + 6) = 3.5

X/Y 1 2 3 4 5 6 0 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111 1 0.0278 0.0278 0.0278 0.0278 0.0278 0.0278 2 0.0278 0.0278 0.0278 0.0278 0.0278 0.0278

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