Question

Prove that when $X \sim N_k(0 . I)$ then $E\left[X_i\right]=0$ and $\operatorname{Cov}\left[X_i, X_j\right]=\delta_{i j}$. Note that you should also justify the existence of the relevant integrals.

    Prove that when $X \sim N_k(0 . I)$ then $E\left[X_i\right]=0$ and $\operatorname{Cov}\left[X_i, X_j\right]=\delta_{i j}$. Note that you should also justify the existence of the relevant integrals.

Approximating integrals via Monte Carlo and deterministic methods

Approximating integrals via Monte Carlo and deterministic methods

Michael Evans, Tim… 1st Edition

Chapter 1, Problem 8

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We are given that \( X \sim N_k(0, I) \), which means \( X \) is a \( k \)-dimensional multivariate normal random vector with mean vector \( 0 \) and covariance matrix \( I \), where \( I \) is the \( k \times k \) identity matrix. We need to prove that the Show more…

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Prove that when $X \sim N_k(0 . I)$ then $E\left[X_i\right]=0$ and $\operatorname{Cov}\left[X_i, X_j\right]=\delta_{i j}$. Note that you should also justify the existence of the relevant integrals.

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