$$ \frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 $$
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 distribution with the following probability density function $$f(x) = \begin{cases} 0.5, & 0 < x < 2 \\ 0.0, & Otherwise \end{cases}$$ Given that the mean of the above probability distribution is 1, the variance (rounded off to two decimal places) is _______________.
- Let $X$ and $Y$ be two independent random variables. $X$ follows $Bernoulli(p = 0.3)$ distribution and $Y$ follows $Normal(\mu = 0, \sigma^2 = 100)$ distribution.
Which of the following options is the variance of $(2X - 1)Y$? - A random variable $X$ has the sample space $\{0,1\}$. The probability $P(X = 0) = 1/4$ and $P(X = 1) = 3/4$.
What is the variance of the random variable?
Hint: $\text{Mean } (\mu) = \sum_{i=1}^{n} x_i p(x_i) ; \text{Variance } (\sigma^2) = \sum_{i=1}^{n} (x_i - \mu)^2 p(x_i)$ The unbiased sample variance for the set of numbers: $S = \{40,45,50,55,60\}$ is_____. (write answer with one decimal place)
A set of 11 (x, y) data points is least-squares fitted to a quadratic polynomial. If the sum of squares of error is 2.4, the variance of error is ________ (round off to 1decimal place).
