George Allen has asked you to analyze his stock portfolio,...

George Allen has asked you to analyze his stock portfolio, which contains 10 shares of stock D and 5 shares of stock C. The joint probability distribution of the stock prices is shown in Table 4.10. Compute the mean and variance for the total value of his stock portfolio. Table 4.10 Joint Probability Distribution for Stock Prices $$ \begin{array}{|ccccc|} \hline {\begin{array}{c} \text { Stock C } \\ \text { Price } \end{array}} & {\text { Stock D Price }} \\ \hline & \$ 40 & \$ 50 & \$ 60 & \$ 70 \\ \$ 45 & 0.00 & 0.00 & 0.05 & 0.20 \\ \$ 50 & 0.05 & 0.00 & 0.05 & 0.10 \\ \$ 55 & 0.10 & 0.05 & 0.00 & 0.05 \\ \$ 60 & 0.20 & 0.10 & 0.05 & 0.00 \\ \hline \end{array} $$

George Allen has asked you to analyze his stock portfolio, which contains 10 shares of stock D and 5 shares of stock C. The joint probability distribution of the stock prices is shown in Table 4.10. Compute the mean and variance for the total value of his stock portfolio.
Table 4.10 Joint Probability Distribution for Stock Prices
$$
\begin{array}{|ccccc|}
\hline {\begin{array}{c}
\text { Stock C } \\
\text { Price }
\end{array}} & {\text { Stock D Price }} \\
\hline & \$ 40 & \$ 50 & \$ 60 & \$ 70 \\
\$ 45 & 0.00 & 0.00 & 0.05 & 0.20 \\
\$ 50 & 0.05 & 0.00 & 0.05 & 0.10 \\
\$ 55 & 0.10 & 0.05 & 0.00 & 0.05 \\
\$ 60 & 0.20 & 0.10 & 0.05 & 0.00 \\
\hline
\end{array}
$$

Key Concepts

Joint Probability Distribution

This is a function that gives the probability that each combination of outcomes for two random variables occurs. It allows us to analyze how two variables behave together, capturing the probability of any pair of outcomes, and is essential for calculating expectations and variances when the variables are not independent.

Expected Value

The expected value, or mean, is the weighted average of all possible outcomes of a random variable, using their probabilities as weights. In the context of a portfolio, it represents the average total value one might expect from the assets, computed by summing the products of outcomes and their probabilities.

Variance

Variance measures the spread of a random variable's values around the mean, indicating the risk or uncertainty associated with the variable. For a portfolio, it helps quantify the variability in total value by taking into account the variability of individual asset prices and their covariances if the assets are not independent.

Covariance

Covariance is a measure of how two random variables change together. In the context of a portfolio with multiple assets, covariance indicates whether the asset prices tend to move in the same or opposite directions, and it is crucial for calculating the overall portfolio variance when the asset prices are jointly distributed.

Linear Combination of Random Variables

A linear combination involves taking a weighted sum of random variables, where the weights are constants. In portfolio analysis, the total portfolio value is a linear combination of the individual asset prices, and the properties such as expectation and variance of the portfolio are derived using the rules that govern linear combinations.

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