Let $X_1, X_2, \dots, X_n$ be independent and identically distributed (i.i.d.) random variables from a normal distribution with mean $\mu$ and variance $\sigma^2$, i.e.,

$X_i \sim N(\mu, \sigma^2), \quad i = 1, 2, \dots, n.$

We are interested in finding the distribution of the sum:

$Y = \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right)^2.$

Step-by-Step Breakdown:

1. Standardization: Each $\frac{X_i - \mu}{\sigma}$ represents a standardized version of $X_i$, which follows a standard normal distribution: $Z_i = \frac{X_i - \mu}{\sigma} \sim N(0, 1).$ So, $Z_i^2$ follows a chi-squared distribution with 1 degree of freedom: $Z_i^2 \sim \chi^2_1.$

2. Sum of Chi-Squared Variables: Since each $Z_i^2$ is independent and follows a $\chi^2_1$ distribution, the sum $Y = \sum_{i=1}^{n} Z_i^2$ is a sum of $n$ independent chi-squared random variables with 1 degree of freedom each.

3. Distribution of $Y$: The sum of $n$ independent chi-squared random variables, each with 1 degree of freedom, follows a chi-squared distribution with $n$ degrees of freedom: $Y \sim \chi^2_n.$

Conclusion:

The sum $Y = \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right)^2$ follows a chi-squared distribution with $n$ degrees of freedom: $Y \sim \chi^2_n.$

Would you like further details or have any questions?

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Here are some related questions to explore further:

1. How do you derive the degrees of freedom for a chi-squared distribution?
2. What are the properties of the chi-squared distribution?
3. Can this result be generalized for non-normal distributions?
4. What is the expectation and variance of a chi-squared distribution?
5. How does this result connect to hypothesis testing, such as in a chi-squared test?

Tip: The chi-squared distribution plays a key role in statistical inference, especially in estimating variance and testing hypotheses.