Question

Consider a random variable $X$ with mean $\mu_X = 0.1$ and variance $\sigma_X^2 = 0.2$. A new random variable $Y = 2X + 1$ is defined. The variance of the random variable $Y$ (rounded off to one decimal place) is _________.

Hint:

Adding a constant to a random variable shifts the mean but does not change the variance. Only the multiplicative factor affects the variance.

Answer 1

Correct Answer: 0.8

To find the variance of the new random variable $Y = 2X + 1$, we use the properties of variance.
Step 1: Recall the variance transformation rule for linear functions:
For any random variable $X$ and constants $a$ and $b$, \[ \mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X). \] The constant term $b$ does not affect the variance because it only shifts the distribution without changing its spread.
Step 2: Identify the constants:
Here, \[ a = 2, \qquad b = 1, \qquad \mathrm{Var}(X) = 0.2. \] Step 3: Apply the formula:
\[ \mathrm{Var}(Y) = (2)^2 \times 0.2 = 4 \times 0.2 = 0.8. \] Step 4: Rounding off:
The value is already at one decimal place.
Thus, the variance of $Y$ is: \[ \boxed{0.8}. \]