CS 70 Discrete Mathematics and Probability Theory DIS 11A Solutions

CS 70 Discrete Mathematics and Probability Theory
Fall 2021 DIS 11A

1 Variance
(a) Let X be a random variable representing the outcome of the roll of one fair 6-sided die. What
is Var (X)?

(b) Let Z be a random variable representing the average of n rolls of a fair die 6-sided die. What
is Var (Z)?

Solution:

(a) Recall that Var (X) = E X 2 − E[X]2 . We can compute each of the individual terms using the
 

definition of expectation:
1 7
E[X] =
(1 + 2 + 3 + 4 + 5 + 6) =
6 2
 2 1 2
E X = 1 + 22 + 32 + 42 + 52 + 62


6
1 91
= (1 + 4 + 9 + 16 + 25 + 36) =
6 6
Now, we plug back into the variance expression:

Var (X) = E X 2 − E[X]2
 
 2
91 7 35
= − =
6 2 12

(b) Because each die roll is independent of the others, we can utilize the fact that for independent
random variables X and Y , Var (X +Y ) = Var (X) + Var (Y ). Let Xi be a random variable




CS 70, Fall 2021, DIS 11A 1

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