Value of a two-stock share bundle

Question

A bundle contains 2 shares of stock X, whose price is $X \sim N(\text{mean } 10,\ \text{variance } 4)$ , and 3 shares of stock Y, whose price is $Y \sim N(\text{mean } 5,\ \text{variance } 1)$ , independent of $X$ . The bundle value is $V = 2X + 3Y$ .

What are the mean and variance of $V$ , and what is its distribution?

Accepted Answer

Scaling a normal by a constant and then adding independent normals keeps everything normal. Two rules combine here: - Mean scales and adds: \mathbb{E}[V] = 2(10) + 3(5) = 20 + 15 = 35. - Variance scales by the square of the constant, then adds: \operatorname{Var}(V) = 2^2(4) + 3^2(1) = 16 + 9 = 25. The key subtlety is that a multiplier a scales the variance by a^2, not by a. So V \sim N(35,\ 25), standard deviation \sqrt{25} = 5. Takeaway. For aX + bY with independence, mean is a\muX + b\muY = 35 and variance is a^2\sigmaX^2 + b^2\sigmaY^2 = 25; multipliers hit the variance squared.

Value of a two-stock share bundle

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