Question

(1) Let us reconsider the profit density function of the previous problem set and answer a few additional questions related to that function: \begin{equation} f(x) = \frac{1}{8}(x+1), \quad -1 < x < 3. \end{equation} (1a) What is the mean or expected profit of your business in a given month, $E(x)$? (1b) Calculate $E(x^2)$ in a similar manner and verify $E(x^2) = 11/3$. Then, verify $Var(x) = 8/9$. (1c) Assuming that profits are independent from month to month, what is the probability that your monthly profits will exceed $1,000 for two consecutive months? Can you generalize your result to quantify the probability of profits exceeding $1,000 for $n$ consecutive months?

          (1) Let us reconsider the profit density function of the previous problem set and answer a few additional questions related to that function:
\begin{equation}
f(x) = \frac{1}{8}(x+1), \quad -1 < x < 3.
\end{equation}
(1a) What is the mean or expected profit of your business in a given month, $E(x)$?
(1b) Calculate $E(x^2)$ in a similar manner and verify $E(x^2) = 11/3$. Then, verify $Var(x) = 8/9$.
(1c) Assuming that profits are independent from month to month, what is the probability that your monthly profits will exceed $1,000 for two consecutive months? Can you generalize your result to quantify the probability of profits exceeding $1,000 for $n$ consecutive months?

(1) Let us reconsider the profit density function of the previous problem set and answer a few additional questions related to that function:

f(x) = (1)/(8)(x+1), -1 < x < 3.

(1a) What is the mean or expected profit of your business in a given month, E(x)?
(1b) Calculate E(x^2) in a similar manner and verify E(x^2) = 11/3. Then, verify Var(x) = 8/9.
(1c) Assuming that profits are independent from month to month, what is the probability that your monthly profits will exceed 1,000 for two consecutive months? Can you generalize your result to quantify the probability of profits exceeding1,000 for n consecutive months?

Added by Stephanie P.

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