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

$1$

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

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)

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