Answer to Question #211901 in Statistics and Probability for mavuyi

Question

Example 2.11 Suppose that the average number of hours a personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation is half an hour.

a.     Find the probability that a personal computer is used for entertainment more than 1 hour per day

b.    Find the probability that a personal computer is used for entertainment more than 4 hours per day

c.     Find the probability that a personal computer is used for entertainment between 1 and 4 hours per day

·        Variance of returns is just one possible risk measure, i.e. different risk measures can be employed when dealing with returns of risky asset

Answer 1

a. Let $X=$ the times for entertainment: $X\sim N(\mu, \sigma^2).$

Given $\mu=2\ h, \sigma=0.5\ h.$

$$P(X>1)=1-P(X\leq 1)$$

$$=1-P(Z\leq \dfrac{1-2}{0.5})=1-P(Z\leq -2)=0.977250$$

b.

$$P(X>4)=1-P(X\leq 4)$$

$$=1-P(Z\leq \dfrac{4-2}{0.5})=1-P(Z\leq 4)=0.000032$$

c.

$$P(1<X<4)=P(X<4)-P(X\leq 1)$$

$$=P(Z< \dfrac{4-2}{0.5})-P(Z\leq \dfrac{1-2}{0.5})$$

$$=P(Z<4)-P(Z\leq -2)$$

$$=0.9999683-0.0227501=0.977218$$