Let X1, . . . , Xn be a random sample from a normal distribution with mean µ and variance σ 2 . What is the distribution of the sum Y = Pn i=1[(Xi − µ)/σ] 2 ?

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.

Let X1, . . . , Xn be a random sample from a normal distribution with mean µ and variance σ². What is the distribution of the sum Y = Σ[(Xi − µ)/σ]²?

Explore the distribution of the sum of standardized normal variables, Y = Σ[(Xi - µ)/σ]², and learn why it follows a chi-squared distribution with n degrees of freedom. This problem covers key concepts in probability and statistics, suitable for undergraduate students.

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