Question

Let \( X \) follow a 10-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Define \( Y = \log_e \sqrt{X^T X} \) and let \( M_Y(t) \) denote the moment generating function of \( Y \) at \( t \), \( t>-10 \). Then \( M_Y(2) \) equals _________ (answer in integer).

Hint:

When dealing with the moment generating function of transformations of multivariate normal distributions, use known formulas and properties of chi-squared distributions to simplify your calculations.

Answer 1

Step 1: Distribution of \( X^T X \) Since \( X \sim \mathcal{N}_{10}(0, I) \), it follows that: \[ X^T X = \sum_{i=1}^{10} X_i^2 \sim \chi^2_{10} \] Step 2: Relation to \( Y \) Given: \[ Y = \frac{1}{2} \log(X^T X) \Rightarrow e^{2Y} = X^T X \] Step 3: Moment Generating Function \[ M_Y(2) = \mathbb{E}[e^{2Y}] = \mathbb{E}[X^T X] \] Since \( X^T X \sim \chi^2_{10} \), and the expected value of a chi-squared distribution with 10 degrees of freedom is 10: \[ \mathbb{E}[X^T X] = 10 \] Final Answer: \[ \boxed{10} \]