Question

In: Statistics and Probability

For Problems 8-10, throw a die twice and let X, Y be the results (each from...

For Problems 8-10, throw a die twice and let X, Y be the results (each from 1 to 6).

Problem 8. Find the expectation and variance of X.

Problem 9. Find the variance of X - 2Y.

Problem 10. Find the expected value of X^2Y.

Expert Solution

Problem 8.

X : Results of throwing a single die

P(X) : Probability of X

Expectation :

Variance : Var(X) = E(X²) - E(X)²

X	P(X)	P(X)	XP(X)	X²P(X)
1	1/6	0.1667	0.1667	0.1667
2	1/6	0.1667	0.3333	0.6667
3	1/6	0.1667	0.5000	1.5000
4	1/6	0.1667	0.6667	2.6667
5	1/6	0.1667	0.8333	4.1667
6	1/6	0.1667	1.0000	6.0000
	Total	1.0000	3.5000	15.1667

Expectation

Expectation = 3.5

Variance : Var(X) = E(X²) - E(X)² = 15.1667 - 3.5² = 2.9167

Problem 9

Variance of X-2Y

The following table provides, All possibles of X,Y and their probability.

X	Y	P(X,Y)
1	1	1/36
1	2	1/36
1	3	1/36
1	4	1/36
1	5	1/36
1	6	1/36
2	1	1/36
2	2	1/36
2	3	1/36
2	4	1/36
2	5	1/36
2	6	1/36
3	1	1/36
3	2	1/36
3	3	1/36
3	4	1/36
3	5	1/36
3	6	1/36
4	1	1/36
4	2	1/36
4	3	1/36
4	4	1/36
4	5	1/36
4	6	1/36
5	1	1/36
5	2	1/36
5	3	1/36
5	4	1/36
5	5	1/36
5	6	1/36
6	1	1/36
6	2	1/36
6	3	1/36
6	4	1/36
6	5	1/36
6	6	1/36

The following table extends the above by calculating X-2Y

X	Y	X-2Y
1	1	-1
1	2	-3
1	3	-5
1	4	-7
1	5	-9
1	6	-11
2	1	0
2	2	-2
2	3	-4
2	4	-6
2	5	-8
2	6	-10
3	1	1
3	2	-1
3	3	-3
3	4	-5
3	5	-7
3	6	-9
4	1	2
4	2	0
4	3	-2
4	4	-4
4	5	-6
4	6	-8
5	1	3
5	2	1
5	3	-1
5	4	-3
5	5	-5
5	6	-7
6	1	4
6	2	2
6	3	0
6	4	-2
6	5	-4
6	6	-6

The above table is sorted on X-2Y, so that same value of X-2Y are together

X	Y	X-2Y
1	6	-11
2	6	-10
1	5	-9
3	6	-9
2	5	-8
4	6	-8
1	4	-7
3	5	-7
5	6	-7
2	4	-6
4	5	-6
6	6	-6
1	3	-5
3	4	-5
5	5	-5
2	3	-4
4	4	-4
6	5	-4
1	2	-3
3	3	-3
5	4	-3
2	2	-2
4	3	-2
6	4	-2
1	1	-1
3	2	-1
5	3	-1
2	1	0
4	2	0
6	3	0
3	1	1
5	2	1
4	1	2
6	2	2
5	1	3
6	1	4

X	Y	X-2Y	X-2Y	P(X-2Y)
1	6	-11	-11	1/36
2	6	-10	-10	1/36
1	5	-9	-9	2/36
3	6	-9
2	5	-8	-8	2/36
4	6	-8
1	4	-7	-7	3/36
3	5	-7
5	6	-7
2	4	-6	-6	3/36
4	5	-6
6	6	-6
1	3	-5	-5	3/36
3	4	-5
5	5	-5
2	3	-4	-4	3/36
4	4	-4
6	5	-4
1	2	-3	-3	3/36
3	3	-3
5	4	-3
2	2	-2	-2	3/36
4	3	-2
6	4	-2
1	1	-1	-1	3/36
3	2	-1
5	3	-1
2	1	0	0	3/36
4	2	0
6	3	0
3	1	1	1	2/36
5	2	1
4	1	2	2	2/36
6	2	2
5	1	3	3	1/36
6	1	4	4	1/36

Z:(X-2Y)	P(Z)=P(X-2Y)	ZP(Z)	Z²P(Z)
-11	0.0278	-0.3056	3.3611
-10	0.0278	-0.2778	2.7778
-9	0.0556	-0.5000	4.5000
-8	0.0556	-0.4444	3.5556
-7	0.0833	-0.5833	4.0833
-6	0.0833	-0.5000	3.0000
-5	0.0833	-0.4167	2.0833
-4	0.0833	-0.3333	1.3333
-3	0.0833	-0.2500	0.7500
-2	0.0833	-0.1667	0.3333
-1	0.0833	-0.0833	0.0833
0	0.0833	0.0000	0.0000
1	0.0556	0.0556	0.0556
2	0.0556	0.1111	0.2222
3	0.0278	0.0833	0.2500
4	0.0278	0.1111	0.4444
	1.0000	-3.5000	26.8333

Let Z= X-2Y

Variance of X-2T i.e Var(Z)

Var(Z) = E(Z²) - E(Z)²

E(Z) = -3.5

E(Z²) = 26.8333

Var(Z) = E(Z²) - E(Z)² = 26.8333 - (-3.5)² = 14.5833

Variance of X-2Y = 14.5833

Problem 10:

Following the above procedure,

W=X²Y

W	P(W)	WP(W)
1	0.0278	0.0278
2	0.0278	0.0556
3	0.0278	0.0833
4	0.0556	0.2222
5	0.0278	0.1389
6	0.0278	0.1667
8	0.0278	0.2222
9	0.0278	0.2500
12	0.0278	0.3333
16	0.0556	0.8889
18	0.0278	0.5000
20	0.0278	0.5556
24	0.0278	0.6667
25	0.0278	0.6944
27	0.0278	0.7500
32	0.0278	0.8889
36	0.0556	2.0000
45	0.0278	1.2500
48	0.0278	1.3333
50	0.0278	1.3889
54	0.0278	1.5000
64	0.0278	1.7778
72	0.0278	2.0000
75	0.0278	2.0833
80	0.0278	2.2222
96	0.0278	2.6667
100	0.0278	2.7778
108	0.0278	3.0000
125	0.0278	3.4722
144	0.0278	4.0000
150	0.0278	4.1667
180	0.0278	5.0000
216	0.0278	6.0000
	1.0000	53.0833

orchestra answered 2 years ago

A fair die is rolled twice. Let X and Y be the smallest and largest, respectively,...

A fair die is rolled twice. Let X and Y be the smallest and largest, respectively, number that appears in the two rolls. (a) Determine the probability mass function of (X, Y). (Write a formula forP(X=i, Y=j)or give a table of values.) (b) Are X and Y independent? (c) Find E(X+Y). (Give your answer as a decimal number.)

Two rolls of a fair die. Let x and y be the results of the two...

Two rolls of a fair die. Let x and y be the results of the two rolls, and let z = x + y. (a) Find P[x = 4, y = 3], P[x > 3, y = 5]. (b) Find P[z = 7], P[z = 5], P[z = 3]. (c) Find the probability that at least one 6 appears given that z = 8. (d) Find the probability that x = 6 given that z > 8. (e) Find the...

1. In the experiment of rolling a balanced die twice. Let X be the minimum of...

1. In the experiment of rolling a balanced die twice. Let X be the minimum of the two numbers obtained and Y be value of the first roll minus the value of the second roll. Determine the probability mass functions and cumulative distribution functions of X and Y, and sketch their graphs please also solve probability for Y

A fair die is rolled twice. Let X be the maximum of the two rolls. Find...

A fair die is rolled twice. Let X be the maximum of the two rolls. Find the distribution of X. Let Y be the minimum of the two rolls. Find the variance of Y.

We throw a die independently four times and let X denote the minimal value rolled. (a)...

We throw a die independently four times and let X denote the minimal value rolled. (a) What is the probability that X ≥ 4. (b) Compute the PMF of X. (c) Determine the mean and variance of X.

2 dice are rolled. Let X be the number on the first die, Y - on...

2 dice are rolled. Let X be the number on the first die, Y - on the second, and Z=X - Y. Find the expectation and standard deviation of Z.

(10 marks) A die is rolled twice. Find the joint probability mass function of X andY...

A die is rolled twice. Find the joint probability mass function of X andY if X denotes the value on the first roll and Y denotes the minimum of the values of the two rolls.

Let X,Y and Z be the following matrices X=[1 10 42 6;5 8 78 23;56 45...

Let X,Y and Z be the following matrices X=[1 10 42 6;5 8 78 23;56 45 9 13;23 22 8 9] Y=[1 2 3;4 10 12;7 21 27] Z=[10 22 5 13] 1. Using single index notation, find the index numbers of the elements in each matrix that contain values of greater than 10. Use MATLAB format

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}. a) Prove or disprove: A ⊆ X b) Prove or disprove: X ⊆ A c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y ) d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

A random variable X is Normally distributed with mean = 75 and = 8. Let Y...

A random variable X is Normally distributed with mean = 75 and = 8. Let Y be a second Normally distributed random variable with mean = 70 and = 12. It is also known that X and Y are independent of one another. Let W be a random variable that is the difference between X and Y (i.e., W = X – Y). What can be said about the distribution of W?