Question

Suppose that a pair of fair dice are to be tossed and let x the random variable denote the sum of the points.

compute the expected value

the variance of X

the standard deviation of x

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} & 1 & 2 &3 & 4 & 5 & 6 \\ \hline 1 & 1+1 & 1+2 & 1+3 & 1+ 4& 1+5 & 1+6 \\ \hdashline 2 & 2+1 & 2+2 & 2+3 & 2+4 & 2+5 & 2+6 \\ \hdashline 3 & 3+1 & 3+2 & 3+3 & 3+4 & 3+5 & 3+6 \\ \hdashline 4 & 4+1 & 4+2 & 4+3 & 4+4 & 4+5 & 4+6 \\ \hdashline 5 & 5+1 & 5+2 & 5+3 & 5+ 4& 5+5 & 5+6 \\ \hdashline 6 & 6+1 & 6+2 & 6+3 & 6+4 & 6+5 & 6+6 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & f & Xf &X^2 & X^2f\\ \hline & 2 & 1/36 & 2/36 & 4 & 4/36 \\ \hdashline & 3 & 2/36 & 6/36 & 9 & 18/36 \\ \hdashline & 4 & 3/36 & 12/36 & 16 & 48/36 \\ \hdashline & 5 & 4/36 & 20/36 & 25 & 100/36 \\ \hdashline & 6 & 5/36 & 30/36 & 36 & 180/36 \\ \hdashline & 7 & 6/36 & 42/36 & 49 & 294/36 \\ \hdashline & 8 & 5/36 & 40/36 & 64 & 320/36 \\ \hdashline & 9 & 4/36 & 36/36 & 81 & 324/36 \\ \hdashline & 10 & 3/36 & 30/36 & 100 & 300/36 \\ \hdashline & 11 & 2/36 & 22/36 & 121 & 242/36 \\ \hdashline & 12 & 1/36 & 12/36 & 144 & 144/36 \\ \hdashline Sum & & 1 & 7 & & 329/6 \end{array}$$

i)

$$E(X)=\sum_ix_ip(x_i)=7$$

ii)

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2$$

$$=329/6-7^2=35/6\approx5.83333$$

iii)

$$\sigma=\sqrt{\sigma^2}=\sqrt{35/6}\approx2.41523$$