Question

given two independent random variables, x with mean 14.6 and standard deviation 0.6, and y with mean 12.1 and standard deviation 0.4, which of the following is a true statement? (a) the mean of x - y is 21.4 (b) the median of x - y is 2.5 (c) the variance of x - y is 0.52 (d) the standard deviation of x - y is 0.2 (e) the standard deviation of x - y is 1.0

Explanation:

Step1: Calculate the mean of \(X - Y\)

For two independent random variables \(X\) and \(Y\), the mean of \(X - Y\) is \(\mu_{X - Y}=\mu_X-\mu_Y\). Given \(\mu_X = 14.6\) and \(\mu_Y=12.1\), then \(\mu_{X - Y}=14.6 - 12.1=2.5\).

Step2: Calculate the variance of \(X - Y\)

For two independent random variables \(X\) and \(Y\), the variance of \(X - Y\) is \(\text{Var}(X - Y)=\text{Var}(X)+\text{Var}(Y)\) (since \(\text{Cov}(X,Y) = 0\) for independent variables). Given \(\sigma_X=0.6\) and \(\sigma_Y = 0.4\), then \(\text{Var}(X)=0.6^{2}=0.36\) and \(\text{Var}(Y)=0.4^{2}=0.16\). So \(\text{Var}(X - Y)=0.36+0.16 = 0.52\).

Step3: Calculate the standard deviation of \(X - Y\)

The standard deviation \(\sigma_{X - Y}=\sqrt{\text{Var}(X - Y)}\). Since \(\text{Var}(X - Y)=0.52\), then \(\sigma_{X - Y}=\sqrt{0.52}\approx0.72
eq0.2\) and \(\sigma_{X - Y}
eq1.0\). Also, for independent random variables, we cannot directly say the median of \(X - Y\) is related to the means in a simple non - distribution - specific way (without knowing the distributions of \(X\) and \(Y\)).

Answer:

C. The variance of \(X - Y\) is \(0.52\)