$$ \frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 $$
Let us denote $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$.
The value of $\frac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2$ is __________ . (Answer in integer)
Variance Calculation Using Sum of Squared Differences
We are given a dataset $\{x_1, x_2, \ldots, x_n\}$ where $n = 100$. The provided equation is:
$ \frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 $We need to find the value of:
$ \frac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2 $where $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$. Let's simplify the double summation term.
Simplifying the Double Summation
The term $\sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2$ can be expanded and simplified. It relates to the sum of squared deviations from the mean ($\sum_{i=1}^{n} (x_i - \bar{x})^2$) by the following identity:
$ \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 2n \sum_{i=1}^{n} (x_i - \bar{x})^2 $Solving for the Target Expression
- Substitute the identity into the given equation: $ \frac{1}{2000} \left( 2n \sum_{i=1}^{n} (x_i - \bar{x})^2 \right) = 99 $
- Substitute the given value $n = 100$: $ \frac{2 \times 100}{2000} \sum_{i=1}^{n} (x_i - \bar{x})^2 = 99 $ $ \frac{200}{2000} \sum_{i=1}^{n} (x_i - \bar{x})^2 = 99 $ $ \frac{1}{10} \sum_{i=1}^{n} (x_i - \bar{x})^2 = 99 $
- Isolate the sum of squared deviations: $ \sum_{i=1}^{n} (x_i - \bar{x})^2 = 99 \times 10 = 990 $
- Calculate the final required value: $ \frac{1}{99} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{99} (990) $ $ = 10 $
The value is an integer, 10.
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 ________________.
- 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.
