I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I followed a certain way to do it, but I am not sure whether what I am doing is mathematically correct.
Assume that we have a multivariate density function $f(x_1,x_2,...,x_N)$. The single variable expectation is calculated simply as $E[x]=\int_{-\infty}^{+\infty}xf(x)dx$. So, intuitively, for the multivariate case, I think about the following vector integration:
$E[<x_1,x_2,...,x_N>^T]=\int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}<x_1,x_2,...,x_N>^Tf(x_1,x_2,...,x_N) dx_1dx_2...dx_N$
What is the mathematically correct way to evaluate this integral? I have my own solution as well but I am not sure that it is valid mathematically.
In the following I give my way of doing the operation:
Since $<x_1,x_2,...,x_N>^Tf(x_1,x_2,...,x_N)dx_1dx_2...dx_N$ is a scalar multiplication of a vector, I write this as: $<x_1f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,x_2f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,...,x_Nf(x_1,x_2,...,x_N)dx_1dx_2...dx_N>^T$.
Now the integration becomes: $E[<x_1,x_2,...,x_N>^T]=\int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}<x_1f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,x_2f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,...,x_Nf(x_1,x_2,...,x_N)dx_1dx_2...dx_N>^T$
As integrating corresponds to an infinite sum so I think about adding uncountable many vectors and I integrate all elements separately as: $E[<x_1,x_2,...,x_N>^T]=<\int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}x_1f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,\int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}x_2f(x_1,x_2,...,x_N)dx_1dx_2...dx_N,...,\int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}x_Nf(x_1,x_2,...,x_N)dx_1dx_2...dx_N>^T$
Since at each $i$th element of the vector all other random variables except $x_i$ are marginalized out, we have: $E[<x_1,x_2,...,x_N>^T]=<\int_{-\infty}^{+\infty}x_1f(x_1)dx_1,\int_{-\infty}^{+\infty}x_2f(x_2)dx_2,...,\int_{-\infty}^{+\infty}x_Nf(x_N)dx_N>^T$
And finally, since each element is the definition of the expected value of a single random variable we obtain the mean vector: $E[<x_1,x_2,...,x_N>^T]=<E[x_1],E[x_2],...,E[x_N]>^T$
My question is simply this: Is my solution approach mathematically correct? If not, what is incorrect here? I am a computer engineer trying to improve
himself in Statistical Machine Learning area, so I don't have a rigorous mathematical background and feel uneasy doing all these calculations.
Thanks in advance.
You should use that $E[<x_1,x_2,...,x_N>]=<\, E[x_1] \ldots E[x_N]\,>$ i.e. a vector of expectations and evaluate each separately e.g.:
$E[x_1]= \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}x_1\,f(x_1,x_2,...,x_N) dx_1dx_2...dx_N$