Question

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

Answer 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.

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

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

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

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

$$=0.4+0.2=0.6$$