Variance Formula for Sum of Random Variables
The general formula for the variance of the sum of two random variables $X$ and $Y$ is:
$ \sigma_{X+Y}^2 = \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y) $
Using the notation $\sigma_X^2 = \text{Var}(X)$ and $\sigma_Y^2 = \text{Var}(Y)$, this can be written as:
$ \sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 + 2\text{Cov}(X,Y) $
Applying Independence Condition
The problem states that the random variables $X$ and $Y$ are statistically independent.
A key property of independent random variables is that their covariance is zero.
$ \text{Cov}(X,Y) = 0 $
Final Variance Calculation
Substitute the covariance of zero into the general variance formula:
$ \sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 + 2(0) $
This simplifies to:
$ \sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 $
Therefore, the variance of the sum of two statistically independent random variables is the sum of their individual variances.
Important Questions from Variance
- 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 ________________.
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.
