Let $T$ be the time that a staff meeting takes. The amount of time for a certain meeting depends on how smoothly it goes. The staff meeting may go smoothly in which case the distribution function of $T$ is $$ F_T(t) = \begin{cases} 0 & \text{for } t < 30 \\ 1 - \frac{(90-t)^2}{3600} & \text{for } 30 \le t \le 90 \\ 1 & \text{for } t > 90 \end{cases} $$ Or the issue of the audit of last year's reserves may come up, in which case the distribution function of $T$ is $$ F_T(t) = \begin{cases} 0 & \text{for } t < 60 \\ \frac{t-60}{30} & \text{for } 60 \le t \le 90 \\ 1 & \text{for } t > 90 \end{cases} $$ The probability that the audit issue will come up is 1/4. Calculate the variance of staff meeting time.
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Let $A$ be the event that the meeting goes smoothly, and $A^c$ be the event that the audit issue comes up. We are given that $P(A^c) = 1/4$, so $P(A) = 1 - 1/4 = 3/4$. The distribution function of $X$ given $A$ is $$ F_{X|A}(x) = \begin{cases} 0 & x < 30 \\ 1 - Show more…
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