Question

1. Consider a rolling two four-sided dice with faces 1, 2, 3 and 4.

(a) Obtain the pmf of X where X is the sum of the two resultant faces

         (b) Suppose the two die were rolled many times. Approximately, what would

would be the average of X?

(c) Calculate the standard deviation of X.

Answer 1

total possible outcomes = 4² = 16

a)

outcome	X	P(X)
(1,1)	2	1/16
(1,2)(2,1)	3	2/16
(1,3)(3,1)(2,2)	4	3/16
(1,4)(4,1)(2,3)(3,2)	5	4/16
(2,4)(4,2)(3,3)	6	3/16
(3,4)(4,3)	7	2/16
(4,4)	8	1/16

b)

X	P(X)	X*P(X)	X² * P(X)
2	0.0625	0.1250	0.2500
3	0.125	0.3750	1.1250
4	0.1875	0.7500	3.0000
5	0.25	1.2500	6.2500
6	0.1875	1.12500	6.75000
7	0.125	0.875	6.125
8	0.063	0.500	4.000

	P(X)	X*P(X)	X² * P(X)
total sum =	1	5	27.50

average=mean = E[X] = Σx*P(X) =            5

c)

E [ X² ] = ΣX² * P(X) =            27.5000
          
variance = E[ X² ] - (E[ X ])² =            2.5000
          
std dev = √(variance) =            1.5811