Question

Let Y be the sum of two fair six-sided die.

(a) Find the PMF of Y.

(b) What is the expected value of Y ?

(c ) What is the standard deviation of Y ?

(d) Interpret the standard deviation you found in the last part in context of the experiment.

(e) Find the CDF of Y.

(f) Use the CDF of Y to find the probability that the sum of the dice will be strictly between six and ten. Answers calculated using the PDF will receive no credit.

Answer 1

Let Y be the sum of two fair six-sided die.

a. The PMF of Y is given by:

Y=y P(Y=y)

2 1/36

3 2/36

4 3/36

5 4/36

6 5/36

7 6/36

8 5/36

9 4/36

10 3/36

11 2/36

12 1/36

b) The expected value of Y is given by:

c) The standard deviation of Y is given by:

Where,

Therefore

d) The standard deviation of Y indicate that the spread of the variable is 2.39 from the mean.

That is the sum of the faces of two fair dice varies by 2.39 from the mean value of 7.

e) The CDF of Y is given by

Y=y P(Y=y) P(Y<=y)

2 1/36 1/36

3 2/36 1/36+2/36=3/36

4 3/36 3/36+3/36=6/36

5 4/36 6/36+/36=10/36

6 5/36 10/36+5/36=15/36

7 6/36 15/36+6/36=21/36

8 5/36 21/36+5/36=26/36

9 4/36 26/36+4/36=30/36

10 3/36 30/36+3/36=33/36

11 2/36 33/36+2/36=35/36

12 1/36 35/36+1/36=36/36=1

e) the probability that the sum of the dice will be strictly between six and ten is given by