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
-
Calculate the variance of \(X\):
Using the given standard deviation \(\sigma_X = 5\):
\( \text{Var}(X) = (\sigma_X)^2 = 5^2 = 25 \)
-
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.
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