Calculate Variance of Transformed Random Variable
We are given a random variable $X$ with mean $\mu_X = 0.1$ and variance $\sigma_X^2 = 0.2$.
A new random variable $Y$ is defined based on $X$ as $Y = 2X + 1$. We need to find the variance of $Y$, denoted as $\sigma_Y^2$.
Variance Properties for Linear Transformation
The variance of a random variable changes in a specific way when it undergoes a linear transformation. For a transformation of the form $Y = aX + b$, where $a$ and $b$ are constants, the variance of $Y$ is related to the variance of $X$ by the following property:
$ \sigma_Y^2 = \text{Var}(aX + b) = a^2 \text{Var}(X) $
In this formula, $a^2$ is the factor by which the variance scales, and the constant $b$ does not influence the variance.
Applying the Variance Formula
For the given transformation $Y = 2X + 1$, we identify the constants: $a = 2$ and $b = 1$.
We are given the variance of $X$ as $\sigma_X^2 = 0.2$.
Now, we substitute these values into the variance property formula:
$ \sigma_Y^2 = a^2 \sigma_X^2 $
$ \sigma_Y^2 = (2)^2 \times 0.2 $
$ \sigma_Y^2 = 4 \times 0.2 $
$ \sigma_Y^2 = 0.8 $
Final Variance Result
The calculated variance of the random variable $Y$ is $0.8$.
The question asks for the variance rounded off to one decimal place. Since $0.8$ already has only one decimal place, no further rounding is needed.
The variance of $Y$ is 0.8.
Important Questions from Variance
- Variance of the sum of two statistically independent random variables $X$ and $Y$, $\sigma_{X+Y}^2$, is
A continuous random variable $x$ has a probability density function given by
$f(x) = e^{-a|x|} \text{ } (-\infty < x < \infty)$
where $a$ is a real constant. The variance of $x$ is __________ (correct up to one decimal place).
- Two yarns have variance of strength as $V_1$ and $V_2$. If $V_1 < V_2$, the variance ratio 'F' would be
People were prohibited ________ their vehicles near the entrance of the main administrative building.
- Analysis of variance (ANOVA) can be used to compare multiple groups of samples. Select the correct option that reflects the principle behind ANOVA.
