Question

Data from a weather forecast station indicate that the annual precipitation in Bandsville is a normal random variable with a mean of 40 inches and a standard deviation of 10 inches. Find the probability that the total precipitation during the next 2 years will exceed 86 inches.

Answer 1

Suppose, random variable $\tiny X_i\left ( i=1,2 \right )$ denotes amount (in inches) of rain in i-th year.

$\tiny \therefore X_i\sim N\left(\mu=40,\sigma^2=10^2\right)$

We define,

$\tiny Y=X_1+X_2$

$\tiny \therefore E\left (Y \right )=E\left (X_1+X_2 \right )=E\left (X_1 \right )+E\left (X_2 \right )=40+40=80$

:: Var (Y) = Var (X1 + X2) = Var (X1) + Var (X2) 102 + 102 = 14.142144

$\tiny \therefore Y\sim N\left(\mu_y=80,\sigma_y^2=14.14214^2\right)$

Required probability is given by

$\tiny P\left ( Y>86 \right )=P\left ( \frac{Y-\mu_y}{\sigma_y}>\frac{86-80}{14.14214} \right )$

$\tiny =P\left ( Z>0.4242639\right )=0.3356867$ [Using R-code '1-pnorm(0.4242639)']