problem 3

(a)

Let

D

be the value of the die roll.

E (D) = \frac{1}{6} (1 + 2 + 3 + 4 + 5 + 6) = 3.5

(b)

Let

T_{4}

be the total sum of the four die rolls, and let

D_{i}

be the value of the

i

-th roll. Note that

T_{4} = D_{1} + D_{2} + D_{3} + D_{4}

. Then, by linearity of expectation,

E (T_{4}) = 4 E (D_{i}) = 4 * 3.5 = 12.2