Suppose a person’s mood changes periodically. The chances of the period being 2 or 4 days are equal and do not affect each other. Suppose the happiness index $h$ in a period is a semicircular curve with the midpoint of the interval as the center.
(a) Draw a sample of happiness index. (b) If $h$ is greater than 3/4, we call him happy. After a long time of observation, what rate of a person’s average happy days is?
I know $E[R_{n}]=E[min\{X,y\}]=\int_{x=0}^{y}Pr\{X>x\}dx$ and $\frac{E[R_{n}]}{\overline{X}}=\frac{1}{\overline{X}}\int_{x=0}^{y}Pr\{X>x\}dx~~WP1$. Obviously, $Pr\{Mean of period width\}=Pr\{\overline{X}\}=2*0.5+4*0.5=3$.
But I don't know how to calculate the value of $\int_{x=0}^{y}Pr\{X>x\}dx$.