Question

Assume that you have a fair 6 sided die with values {1, 2, 3, 4, 5, 6} and a fair 12 sided die with values {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. A discrete random variable is generated by rolling the two dice and adding the numerical results together.

(a) Create a probability mass function that captures the probability of all possible values of this random variable. You may use R or draw the pmf on paper.

(b) Find the expected value of this discrete random variable. Make sure to show your work in calculating this.

(c) Find the variance of this discrete random variable. Make sure to show your work in calculating this.

please answer all

Answer 1

Assume that you have a fair 6 sided die with values {1, 2, 3, 4, 5, 6} and a fair 12 sided die with values {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. A discrete random variable is generated by rolling the two dice and adding the numerical results together.

(a) Create a probability mass function that captures the probability of all possible values of this random variable. You may use R or draw the pmf on paper.

Sample space:

Number on dice	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12
7	8	9	10	11	12	13
8	9	10	11	12	13	14
9	10	11	12	13	14	15
10	11	12	13	14	15	16
11	12	13	14	15	16	17
12	13	14	15	16	17	18

PMF:

x	frequency	p(x)
2	1	0.01389
3	2	0.02778
4	3	0.04167
5	4	0.05556
6	5	0.06944
7	6	0.08333
8	6	0.08333
9	6	0.08333
10	6	0.08333
11	6	0.08333
12	6	0.08333
13	6	0.08333
14	5	0.06944
15	4	0.05556
16	3	0.04167
17	2	0.02778
18	1	0.01389
Total	72	1.00000

(b) Find the expected value of this discrete random variable. Make sure to show your work in calculating this.

X	P(X)	x*p(x)	(x-mean)^2*p(x)
2	0.01389	0.02778	0.88894
3	0.02778	0.08334	1.36118
4	0.04167	0.16668	1.50007
5	0.05556	0.27780	1.38894
6	0.06944	0.41664	1.11098
7	0.08333	0.58331	0.74992
8	0.08333	0.66664	0.33329
9	0.08333	0.74997	0.08331
10	0.08333	0.83330	0.00000
11	0.08333	0.91663	0.08335
12	0.08333	0.99996	0.33335
13	0.08333	1.08329	0.75002
14	0.06944	0.97216	1.11110
15	0.05556	0.83340	1.38906
16	0.04167	0.66672	1.50017
17	0.02778	0.47226	1.36126
18	0.01389	0.25002	0.88898
Total	1.000	9.9999	14.8339

= 9.9999

(c) Find the variance of this discrete random variable. Make sure to show your work in calculating this.

=14.8339