Mean and Variance of 2X - Y

Solution & Explanation

When dealing with linear combinations of independent normal random variables, it's essential to understand how both the mean and variance transform. Let's break down the problem:

Given:

• X is a random variable following a normal distribution: $X \sim N(3, 4)$
  • Mean of $X$: $E(X) = 3$
  • Variance of $X$: $Var(X) = 4$
• Y is another independent random variable: $Y \sim N(1, 4)$
  • Mean of $Y$: $E(Y) = 1$
  • Variance of $Y$: $Var(Y) = 4$

We're tasked with finding the mean and variance of the expression $2X - Y$.

Mean Calculation:

The mean of a linear combination of random variables is the linear combination of their means:

$E(2X - Y) = 2E(X) - E(Y)$

Substituting the known values:

$E(2X - Y) = 2(3) - 1 = 6 - 1 = 5$

Variance Calculation:

The variance of a linear combination of independent random variables is calculated by the formula:

$Var(aX + bY) = a^2Var(X) + b^2Var(Y)$

Since $X$ and $Y$ are independent, their covariance is zero, i.e., $Cov(X, Y) = 0$. Thus, the variance of $2X - Y$ is:

$Var(2X - Y) = 2^2Var(X) + (-1)^2Var(Y)$

Plugging in the known variances:

$Var(2X - Y) = 4 \times 4 + 1 \times 4 = 16 + 4 = 20$

Conclusion:

• The mean of the distribution of $2X - Y$ is $5$.
• The variance of the distribution of $2X - Y$ is $20$.

These calculations confirm that the linear combination of independent normal random variables results in another normal distribution, where the mean and variance are derived from the linear properties of expectation and variance.