Show that E (x; – x)² = E-1 x? – nữ², hence show the following equivalent formula for the sample variance E, (®; – ¤)² _ E, ? – nã? п — 1 п — 1
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Variance Derivation Exercise
**Problem Statement:**
2. Show that \(\sum_{i=1}^{n} (x_i - \bar{x})^2 = \sum_{i=1}^{n} x_i^2 - n\bar{x}^2\), hence show the following equivalent formula for the sample variance:
\[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} = \frac{\sum_{i=1}^{n} x_i^2 - n\bar{x}^2}{n - 1} \]
### Explanation:
This problem involves proving an identity related to the sum of squared deviations from the mean and then using it to derive an equivalent formula for the sample variance.
1. **Sum of Squared Deviations:**
- You need to show that the sum of the squared deviations from the mean \((x_i - \bar{x})^2\) equals the total sum of squares minus \(n\) times the square of the mean.
- Begin by expanding the squared term and simplifying it to arrive at the given identity.
2. **Sample Variance Formula:**
- Once the identity is proved, it can be used to derive an equivalent form of the sample variance.
- The sample variance \(s^2\) is typically defined as the sum of squared deviations divided by \(n - 1\) (where \(n\) is the sample size).
- By substituting the identity into this formula, you can express the variance in terms of the sum of the squares of the observations and the mean.
### Mathematical Representation:
- **Sum of Squared Deviations Identity:**
\[
\sum_{i=1}^{n} (x_i - \bar{x})^2 = \sum_{i=1}^{n} x_i^2 - n\bar{x}^2
\]
- **Sample Variance Formula:**
\[
s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1} = \frac{\sum_{i=1}^{n} x_i^2 - n\bar{x}^2}{n - 1}
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