# A review of random vectors

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## Outline

• The following topics will be covered in this lecture:
• Multiple random variables
• CDF and PDF of multiple variables
• Marginals
• The ensemble matrix
• The expected value
• The ensemble mean

## Introducing multiple random variables

• We will now introduce the basic tools of statistics and probability theory for multivariate analysis;

• we will be studying the relations between $$N_x>1$$ total RVs that will often covary together in their conditional probabilities.
• Some notions like the expected / center of mass will translate directly over linear combinations of RVs.

• Recall, for RVs $$X,Y$$ and a constant scalars $$a,b$$ we have $\mathbb{E}\left[ a X + b Y\right] = a \mathbb{E}\left[X\right] + b \mathbb{E}\left[Y\right]$ by the linearity of the expectation.

• Recall the vector notation $$\pmb{x} \in \mathbb{R}^{N_x}$$, matrix notation $$\mathbf{A} \in \mathbb{R}^{N_x \times N_x}$$, and matrix-slice notation $$\mathbf{A}^j \in \mathbb{R}^{N_x}$$ where

\begin{align} \pmb{x} := \begin{pmatrix} x_1 \\ \vdots \\ x_{N_x} \end{pmatrix} & & \mathbf{A} := \begin{pmatrix} a_{1,1} & \cdots & a_{1, N_x} \\ \vdots & \ddots & \vdots \\ a_{N_x,1} & \cdots & a_{N_x, N_x} \end{pmatrix} & & \mathbf{A}^j := \begin{pmatrix} a_{1,j} \\ \vdots \\ a_{N_x, j} \end{pmatrix} \end{align}

• The linearity of the expectation then extends to random vectors $$\pmb{x}, \pmb{y}$$ and constant matrices $$\mathbf{A},\mathbf{B}$$, we can write $\mathbb{E}\left[\mathbf{A}\pmb{x} + \mathbf{B}\pmb{y} \right] = \mathbf{A}\mathbb{E}\left[ \pmb{x}\right] + \mathbf{B}\mathbb{E}\left[\pmb{y}\right].$

• While the concept of the center of mass has a direct generalization to vectors, we will need to make some additional considerations when we measure the spread of RVs and how they relate to others.

• The extension of the second centered moment has the generalization to the rotational inertia tensor.

## The cumulative distribution function

• We will begin our consideration in $$N_x=2$$ dimensions, as all properties described in the following will extend (with minor modifications) to arbitrarily large but finite $$N_x$$.

• Let the random vector $$\pmb{X}$$ be defined as

\begin{align} \pmb{X} = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \end{align} where each of the above components $$X_i$$ is a RV.

• We can define the cumulative distribution function in a similar way to the definition in one variable.

• Let $$x_1,x_2$$ be two fixed real values forming a constant vector as

$\pmb{x} = \begin{pmatrix} x_1 \\ x_2\end{pmatrix}.$

• Define the comparison operator between two vectors $$\pmb{y}, \pmb{x}$$ as

$\pmb{y} \leq \pmb{x} \Leftrightarrow y_i \leq x_i \text{ for each and every }i$

Multivariate cumulative distribution function
The cumulative distribution function $$P$$, describing the probability of realizations of $$\pmb{X}$$, is defined $P(\pmb{x}) = \mathcal{P}(\pmb{X}\leq \pmb{x} ) = \mathcal{P}(X_i \leq x_i \quad \forall i=1,\cdots,N_x).$

### The joint probability density function

• Recall that the CDF

\begin{align} P:\mathbb{R}^2 & \rightarrow [0,1] \\ \pmb{x} & \rightarrow \mathcal{P}(\pmb{X}\leq \pmb{x}) \end{align} is a function of the variables $$(x_1,x_2)$$.

• Suppose then that $$P$$ has continuous second partial derivatives in $$\partial_{x_1} \partial_{x_2}P = \partial_{x_2}\partial_{x_1}P$$;

• this implies we can arbitrarily exchange the order of differentiation or integration.
Multivariate probability density function
Let $$P\in \mathcal{C}^2\left(\mathbb{R}^2\right)$$ , then the probability density function $$p$$ is defined as \begin{align} p:\mathbb{R}^2 & \rightarrow \mathbb{R}\\ \pmb{x} &\rightarrow \partial_{x_1}\partial_{x_2}P(\pmb{x}) \end{align}
• In the above definition, we have constructed the density function in the same way as in one variable;

• specifically, in the case where $$p$$ itself is differentiable, we have defined the CDF as the anti-derivative of the density:

\begin{align} P(\pmb{x}) = \int_{-\infty}^{x_1} \int_{-\infty}^{x_2} p(s_1, s_2) \mathrm{d}s_1 \mathrm{d}s_2. \end{align}

### The joint probability density function

• Recall that the relationship between the CDF and the PDF as

\begin{align} P(\pmb{x}) = \int_{-\infty}^{x_1} \int_{-\infty}^{x_2} p(s_1, s_2) \mathrm{d}s_1 \mathrm{d}s_2; \end{align}

• this implies that

\begin{align} P(- \infty < \pmb{X} < \infty) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(s_1, s_2) \mathrm{d}s_1 \mathrm{d}s_2 = 1 \end{align}

• If we define this over $$N_x\geq 2$$ variables, all of the above extends identically when $$P$$ has derivatives defined in arbitrary arrangements of the $$N_x$$-th partial derivatives in each $$\partial_{x_i}$$.

• Specifically, this requires that, for all permutations $$\psi:(1, \cdots, N_x) \rightarrow (\psi(1), \cdots ,\psi(N_x))$$,

\begin{align} \partial_{x_1} \cdots \partial_{x_{N_x}}P = \partial_{x_{\psi(1)}} \cdots \partial_{x_{\psi(N_x)}}P \end{align}

• We then construct $$p$$ as the $$N_x$$-th partial derivative of $$P$$ in all univariate partial derivatives.
• Using the integral relationship, we can similarly show

\begin{align} P(\pmb{x}) = \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_{N_x}} p(s_1, \cdots, s_{N_x}) \mathrm{d}s_1\cdots \mathrm{d}s_{N_x}. \end{align}

### The joint probability density function

• Particularly, we will again view the density function like the curve of the univariate case, but for two variables we see this as a surface above the $$x_1,x_2$$ plane. Courtesy of: F.M. Dekking, et al. A Modern Introduction to Probability and Statistics. Springer Science & Business Media, 2005.

• To the right, we see the multivariate Gaussian bell surface that defines the multivariate normal distribution in two variables.
• In one variable, the probability $$\mathcal{P}(X_1 \leq x_1)$$ was associated to the area under the curve, computed by the integral of the density.