Standard multivariate normal distribution

Dependencies:

1. Random variable
2. Normal distribution
3. Independence of random variables (incomplete)
4. Covariance matrix
5. Identity matrix

$\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Var}{\operatorname{Var}}$ $\newcommand{\Cov}{\operatorname{Cov}}$

Let $Z = [Z_1, Z_2, \ldots, Z_d]$ be a sequence of random variables. $Z$ is said to follow the $d$-dimensional standard multivariate normal distribution (denoted by $Z \sim \mathcal{N}(0, 1)^d$) iff all $Z_i$ are independent and each $Z_i \sim \mathcal{N}(0, 1)$.

$Z$ has a probability density function given by \[ f_Z(z) = \prod_{i=1}^d \phi(z_i) = \frac{1}{\sqrt{(2\pi)^d}} \exp\left( -\frac{1}{2} \|z\|^2 \right) \] This is because the joint probability density of independent random variables is the product of the probability densities of the components.

Note that the probability density at $z$ depends only on the distance of $z$ from the origin; the individual components of $z$ don't matter. This is why the standard multivariate normal distribution is also called the spherical gaussian distribution.

$\E(Z) = 0$. When $i \neq j$, $\Var(Z)_{i, j} = \Cov(Z_i, Z_j) = 0$. $\Var(Z)_{i, i} = \Var(Z_i) = 1$. Therefore, $\Var(Z) = I$.

Dependency for:

1. Standard normal random vector on vector space
2. General multivariate normal distribution

Info:

• Depth: 11
• Number of transitive dependencies: 36

Transitive dependencies:

1. /analysis/exponential-grows-superpolynomially
2. /analysis/integration-by-parts
3. /measure-theory/linearity-of-lebesgue-integral
4. /measure-theory/lebesgue-integral
5. /sets-and-relations/de-morgan-laws
6. /sets-and-relations/countable-set
7. /analysis/topological-space
8. Gaussian integral (incomplete)
9. Group
10. Ring
11. Field
12. Vector Space
13. Semiring
14. Matrix
15. Identity matrix
16. σ-algebra
17. σ-algebra is closed under countable intersections
18. Measure
19. Probability
20. Conditional probability (incomplete)
21. Independence of events
22. Independence of composite events
23. Generated σ-algebra
24. Measurable function
25. Borel algebra
26. Generators of the real Borel algebra (incomplete)
27. Random variable
28. Expected value of a random variable
29. Linearity of expectation
30. Covariance of 2 random variables
31. Variance of a random variable
32. Normal distribution
33. Linearity of expectation for matrices
34. Cross-covariance matrix
35. Covariance matrix
36. Independence of random variables (incomplete)