Question

Let W be a random variable that takes values 1 to 6 with equal probability.

a) Write the PMF. 


b) Calculate the mean and variance of W and z=10 W.

Answer 1

W be a random variable that takes values 1 to 6 with equal probability

so, each will have probability of 1/6 = 0.1667

a)

so, PMF will be

W	P(W)
1	0.1667
2	0.1667
3	0.1667
4	0.1667
5	0.1667
6	0.1667

b)

W	P(W)	W*P(W)	W² * P(W)
1	0.1667	0.167	0.167
2	0.1667	0.333	0.667
3	0.1667	0.500	1.500
4	0.1667	0.667	2.667
5	0.1667	0.833	4.167
6	0.1667	1.000	6.000

	P(W)	W*P(W)	W² * P(W)
total sum =	1	3.5	15.16667

mean = E[W] = ΣW*P(W) =            3.5
          
E [ W² ] = ΣW² * P(W) =            15.1667
          
variance = E[ W² ] - (E[ W ])² =            2.91667

-----------------------------------------

Z=10W

Z	P(Z)	Z*P(Z)	Z² * P(Z)
10	0.1667	1.667	16.667
20	0.1667	3.333	66.667
30	0.1667	5.000	150.000
40	0.1667	6.667	266.667
50	0.1667	8.333	416.667
60	0.1667	10.000	600.000

	P(Z)	Z*P(Z)	Z² * P(Z)
total sum =	1	35	1516.667

mean = E[Z] = ΣZ*P(Z) =            35
          
E [ Z² ] = ΣZ² * P(Z) =            1516.666667
          
variance = E[ Z² ] - (E[ Z ])² =            291.6666667