The number of patients seen in the Emergency Room in any given hour is a random variable represented by x. The probability distribution for x is : X 10 11 12 13 14 P(X) 0.4 0.2 0.2 0.1 0.1

1.

$E(X)=\sum _ix_ip(x_i)=10(0.4)+11(0.2)$

$+12(0.2)+13(0.1)+14(0.1)=11.3$

$E(X^2)=\sum _ix_i^2p(x_i)=10^2(0.4)+11^2(0.2)$

$+12^2(0.2)+13^2(0.1)+14^2(0.1)=129.5$

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

$=129.5-(11.3)^2=1.81$

$\sigma=\sqrt{\sigma^2}=\sqrt{1.81}=1.3454$

2.

$P(X=14)=0.1$$P(X\ge 14)=0.1$

$P(X\ge 12)=P(X=12)+P(X=13)$

$+P(X=14)=0.2+0.1+0.1=0.4$

$P(X\le 11)=P(X=10)+P(X=11)$

$=0.4+0.2=0.6$