Linear Combination of Independent Normal Variables

Solution & Explanation

To find the mean and variance of the expression $2X - Y$, where $X$ and $Y$ are independent normal random variables, we can utilize properties of linear combinations of random variables.

Given:

• $X \sim N(3, 4)$

  • Mean of $X$: $\mu_X = 3$
  • Variance of $X$: $\sigma_X^2 = 4$

• $Y \sim N(1, 4)$

  • Mean of $Y$: $\mu_Y = 1$
  • Variance of $Y$: $\sigma_Y^2 = 4$

Expression:

• We need to find the mean and variance of $2X - Y$.

Mean of $2X - Y$:

The mean of a linear combination $aX + bY$ is given by: $E(aX + bY) = aE(X) + bE(Y)$

For $2X - Y$:

• $a = 2$, $b = -1$
• $E(2X - Y) = 2E(X) - E(Y)$
• $E(2X - Y) = 2(3) - 1 = 6 - 1 = 5$

So, the mean of $2X - Y$ is 5.

Variance of $2X - Y$:

The variance of a linear combination $aX + bY$ is given by: $Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2ab \cdot Cov(X, Y)$

Since $X$ and $Y$ are independent, $Cov(X, Y) = 0$, thus: $Var(2X - Y) = 2^2Var(X) + (-1)^2Var(Y)$

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

So, the variance of $2X - Y$ is 20.

Conclusion:

The expression $2X - Y$ follows a normal distribution with:

• Mean: 5
• Variance: 20

Thus, $2X - Y \sim N(5, 20)$.

This analysis demonstrates how to utilize properties of linear combinations of independent normal variables to determine the resulting distribution's parameters.