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