What is meant by "dot product between random variables?"

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I was having a discussion with a colleague today about correlation coefficients, and I was told that correlation coefficient between 2 random variables $X$ and $Y$ is proportional to the dot product of the two random variables.

I asked him what he means by this, and I was told that you can view random variables as vectors. I don't think I agree with that, but I don't have a sufficient background to really argue my point, but now I want to revisit this.

How can a random variable be viewed a vector? What is meant by dot product between 2 random variables -- is this actually formal terminology or something loosely used?

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4 Answers

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The space $L^0(\Omega)$ of all random variables on a fixed sample space $\Omega$ is a vector space - the (outcome-wise) sum of two random variables is a random variable, and a scalar multiple of a random variable is again a random variable. So in that sense, random variables can be viewed as "vectors" because they are the elements of a vector space.

By "dot product" they likely mean the $L^2$ inner product, defined by $\langle X, Y \rangle = E[XY]$. This obeys the same basic algebraic properties as the ordinary Euclidean dot product: bilinear (with respect to the addition and scalar multiplication described above), symmetric, positive definite. Strictly speaking, this inner product doesn't necessarily live on $L^0(\Omega)$, but rather on the vector subspace $L^2(\Omega) \subset L^0(\Omega)$ consisting of random variables with finite second moment.

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For two joint discrete variables, the expectation of their product is a weighted dot product of their value vectors (all diagonal values are positive making the diagonal matrix positive definite):

$$ \mathbf{E}[XY] = \sum_{i=1}^n p_i x_i y_i = (x_1,...,x_n) \begin{pmatrix} p_1 & ... & 0\\ \vdots & \ddots & \vdots \\ 0 & ...& p_n \end{pmatrix} (y_1,...,y_n)^T$$

Here, $(X,Y)$ has $n$ possible realizations $(x_i, y_i)$ with probabilities $p_i$, $i=1,...,n$.

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Suppose you have a collection of $n$ samples of dependent (in general) variables $X$ and $Y$: $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$

Then we can view this collection of $n$ samples as a pair of vectors in $\mathbb{R}^n$: $(x_1, x_2, \ldots, x_n)$ and $(y_1, y_2, \ldots, y_n)$.

Then what your colleague is saying is that we can view correlation between $X$ and $Y$ as a kind of normalized inner product between these two vectors.

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A multivariate random variable can be considered as a random vector.

But correlation between two such random vectors (or more precisely, cross-correlation) would typically produce a matrix rather than a scalar value

My guess is that you may have been discussing two univariate random variables, say $X$ and $Y$, and calculating the sample correlation between them. If the sample size is $n$ then you could regard the two samples as random vectors $\mathbf{X}=(X_1,X_2,\ldots,X_n)$ and $\mathbf{Y}=(Y_1,Y_2,\ldots,Y_n)$. The sample correlation coefficient would then be $$\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}}$$ but you could calculate this using dot products and scalar arithmetic with the vector $\mathbf{1}_n$ of $n$ ones, with $$\frac{\mathbf X \cdot \mathbf Y - n(\mathbf X \cdot \mathbf 1_n)(\mathbf Y \cdot \mathbf 1_n) }{\sqrt{(\mathbf X \cdot \mathbf X - n(\mathbf X \cdot \mathbf 1_n)^2)(\mathbf Y \cdot \mathbf Y - n(\mathbf Y \cdot \mathbf 1_n)^2)}}$$

If you know that the expected values of $X$ and $Y$ are zero then you can use$$\frac{\sum\limits_{i=1}^n x_i y_i}{\sqrt{\sum\limits_{i=1}^n x_i^2 \sum\limits_{i=1}^n y_i^2}} \text{ or }\frac{\mathbf X \cdot \mathbf Y }{\sqrt{(\mathbf X \cdot \mathbf X)(\mathbf Y \cdot \mathbf Y )}}$$and in this sense you might be stretching things and the correlation is proportional to the sample covariance $\mathbf X \cdot \mathbf Y$

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