To show that the sum of squares $\sum_{i=1}^{n} (X_i - \bar{X})^2$ can be rewritten in the specified forms, we’ll apply formulas (1.14.6) and (1.9.1). For this explanation, let's denote:

• $X_i$ as the $i$-th observation,
• $\bar{X}$ as the mean of $X_i$ values, i.e., $\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$,
• $n$ as the number of observations.

Part (a): Show that

$\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - n \bar{X}^2$

Step-by-Step Solution:

1. Expand the left-hand side: Start with the definition of the sum of squared deviations: $\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} \left( X_i^2 - 2 X_i \bar{X} + \bar{X}^2 \right).$

2. Separate the terms: Distribute the summation: $= \sum_{i=1}^{n} X_i^2 - 2 \sum_{i=1}^{n} X_i \bar{X} + \sum_{i=1}^{n} \bar{X}^2.$

3. Simplify each term:

   • The first term, $\sum_{i=1}^{n} X_i^2$, remains as it is.
   • The second term becomes $-2 \bar{X} \sum_{i=1}^{n} X_i$. Since $\sum_{i=1}^{n} X_i = n \bar{X}$, this term simplifies to $-2 \bar{X} \cdot n \bar{X} = -2 n \bar{X}^2$.
   • The third term, $\sum_{i=1}^{n} \bar{X}^2$, simplifies to $n \bar{X}^2$ because $\bar{X}^2$ is constant with respect to $i$.

4. Combine terms: $\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - 2 n \bar{X}^2 + n \bar{X}^2 = \sum_{i=1}^{n} X_i^2 - n \bar{X}^2.$

This proves part (a).

---

Part (c): Show that

$\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - \bar{X} \sum_{i=1}^{n} X_i.$

1. Use the result from part (a): From part (a), we know that $\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - n \bar{X}^2.$

2. Substitute $n \bar{X}$ with $\sum_{i=1}^{n} X_i$: Recall that $\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$, so $n \bar{X} = \sum_{i=1}^{n} X_i$.

3. Rewrite $n \bar{X}^2$: $n \bar{X}^2 = \bar{X} \sum_{i=1}^{n} X_i.$

4. Substitute into the formula: $\sum_{i=1}^{n} (X_i - \bar{X})^2 = \sum_{i=1}^{n} X_i^2 - \bar{X} \sum_{i=1}^{n} X_i.$

This completes the proof for part (c).

---

Let me know if you’d like further details or have any questions about these derivations.

---

Relative Questions:

1. What is the geometric interpretation of $\sum_{i=1}^{n} (X_i - \bar{X})^2$ in statistics?
2. How can these rewritten forms be useful for computational efficiency?
3. What is the meaning of $\bar{X}$ in terms of probability if $X_i$ represents a sample?
4. How does this approach change if $X_i$ values are weighted?
5. What role does the variance play in relation to these forms?

Tip: Remember, expanding and simplifying terms step-by-step helps avoid mistakes when dealing with summation formulas.