GATE Overflow Test Series | Probability | Test 1 | Question: 10

Question

The expected value of the product when two dice are rolled is _____

Comments

Sample space for both dices = $\left \{ 1,2,3,4,5,6\left. \right \} \right.$

$E(x) = \sum xP_{x}$   ($x$ is expected value of product)

(Every element of sample space of first dice is mutiplied with every element of sample space of second dice)

E(x) = probability of 1 multiplicaion [(1 is mutiplied with each element in 2^nd  dice) + (2 is mutiplied with each element in 2^nd dice)……………. +(6 is mutiplied with each element in 2^nd dice)]

$E(x) = 1/6 * 1/6* [1*(6*7)/2 + 2*(6*7)/2 +3*(6*7)/2 +…...+6*(6*7)/2]$

$E(x) = 1/36 * (6*7)/2 * [1+2+3…..+6]$

$E(x) = 49/4 = 12.25$

Answer 1

The possibles values of product are $1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36.$

Expected value, $E = \displaystyle \sum_i iP(i) = 1 \times \frac{1}{36} + 2 \times \frac{2}{36} + 3 \times \frac{2}{36} + 4 \times \frac{3}{36}+ 5 \times \frac{2}{36}+ 6 \times \frac{4}{36}+ 8 \times \frac{2}{36}+ 9 \times \frac{1}{36}+ 10 \times \frac{2}{36}+ 12 \times \frac{4}{36}+ 15 \times \frac{2}{36}+ 16 \times \frac{1}{36}+ 18 \times \frac{2}{36}+ 20 \times \frac{2}{36}+ 24 \times \frac{2}{36}+ 25 \times \frac{1}{36}+ 30 \times \frac{2}{36}+ 36 \times \frac{1}{36}$

$\quad = \dfrac{1+4+6+12+10+24+16+9+20+48+30+16+36+40+48+25+60+36}{36}$

$\quad= \dfrac{441}{36}=12.25  $

Comments

If Random variable X and Y are independent then E[X*Y] = E[x] * E[y] = 3.5 * 3.5 = 12.25

Answer 2

The expectation of a product of independent random variables is equal to the product of their expectations.

Let X1 and X2 be scores by first and second die respectively. Note that X1and X2 are independent. Then

E(X1X2)=E(X1)E(X2)

E(X1)= 1+2+3+4+5+6 / 6

         = 21/6 = 7/2 = 3.5  = E(X2)

E(X1X2)= (3.5)(3.5) = 12.25

I hope my answer helps you a lot.