To address part 1 of the problem, we need to prove that the quantity $(X - \mu)^T \Sigma^{-1} (X - \mu)$ is distributed as a chi-squared distribution with $p$ degrees of freedom, given that $X$ is distributed as $N_p(\mu, \Sigma)$, where $\mu$ is the mean vector and $\Sigma$ is the covariance matrix.

Proof:

Step 1: Definition and Setup

• Let $X$ be a random vector following a multivariate normal distribution: $X \sim N_p(\mu, \Sigma)$ This means: $X \text{ has mean vector } \mu \text{ and covariance matrix } \Sigma.$

Step 2: Standardization

• Define the standardized random vector $Z$ as: $Z = \Sigma^{-\frac{1}{2}} (X - \mu)$ Here, $Z$ is a random vector with a standard normal distribution $N_p(0, I)$, where $I$ is the identity matrix of size $p$. This is because: $Z \sim N_p(0, I)$ where $Z$ has mean $0$ and covariance matrix $I$.

Step 3: Expression of the Quadratic Form

• Consider the quadratic form: $(X - \mu)^T \Sigma^{-1} (X - \mu)$ Substitute $Z$ into the expression: [ (X - \mu)^T \Sigma^{-1} (X - \mu) = \left[\Sigma^{\frac{1}{2}}Z\right]^T \Sigma^{-1} \left[\Sigma^{\frac{1}{2}}Z\right] ] Simplifying: $= Z^T \Sigma^{\frac{1}{2}T} \Sigma^{-1} \Sigma^{\frac{1}{2}} Z = Z^T Z$

Step 4: Distribution of the Quadratic Form

• Since $Z \sim N_p(0, I)$, the sum of the squares of $p$ independent standard normal variables $Z_1^2 + Z_2^2 + \cdots + Z_p^2$ follows a chi-squared distribution with $p$ degrees of freedom: $Z^T Z = Z_1^2 + Z_2^2 + \cdots + Z_p^2 \sim \chi^2_p$

Conclusion:

• Therefore, $(X - \mu)^T \Sigma^{-1} (X - \mu)$ follows a chi-squared distribution with $p$ degrees of freedom: $(X - \mu)^T \Sigma^{-1} (X - \mu) \sim \chi^2_p$

This completes the proof.

Do you have any questions on this proof? Would you like to proceed to part 2 or any other section?

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Related Questions:

1. How does the interpretation of $(X - \mu)^T \Sigma^{-1} (X - \mu)$ change when the covariance matrix $\Sigma$ is not diagonal?
2. What role does the covariance matrix $\Sigma$ play in shaping the distribution of the quadratic form?
3. Can you provide a practical example where this chi-squared distribution would be useful in real-world data analysis?
4. How would you modify this proof if $X$ followed a non-normal distribution?
5. Why is it important that the random vector $Z$ is standardized in this proof?

Tip:

When proving distribution properties, it's often useful to standardize the variables involved. This reduces the problem to working with standard distributions, making it easier to apply known results.