Variance Calculation for Transformed Random Variable

This question requires calculating the variance of a transformed random variable $Y$ based on the properties of the original random variable $X$. We are given the mean and standard deviation of $X$.

Given Information:

• Random variable $X$ has mean $\mu_X = 3$.
• Random variable $X$ has standard deviation $\sigma_X = 5$.
• The transformation is $Y = 2X - 5$.

Key Variance Properties

Recall the properties of variance:

• The variance is the square of the standard deviation: $\text{Var}(X) = (\sigma_X)^2$.
• For a linear transformation $aX + b$, the variance is $\text{Var}(aX + b) = a^2 \text{Var}(X)$. The constant term $b$ does not affect the variance.

Step-by-Step Solution

1. Calculate the variance of $X$:

   Using the given standard deviation $\sigma_X = 5$:

   $ \text{Var}(X) = (\sigma_X)^2 = 5^2 = 25 $

2. Calculate the variance of $Y$:

   For the transformation $Y = 2X - 5$, we have $a=2$ and $b=-5$. Apply the variance property:

   $ \text{Var}(Y) = \text{Var}(2X - 5) = 2^2 \times \text{Var}(X) $

   Substitute the calculated $\text{Var}(X)$:

   $ \text{Var}(Y) = 4 \times 25 = 100 $

Final Answer

The variance of the random variable $Y = 2X - 5$ is 100.