I am interested in finding a conditional expectation of a multivariate Normal distribution. For example, let $X$ and $Y$ denote two random variables with a joint normal distribution.

Let $X \sim N(\mu_x, \sigma_x)$, $Y\sim N(\mu_y, \sigma_y)$ and $Cov(X,Y) = \sigma^{2}_{x}$

I want to compute for example $E(X|Y)$. While I know how to get this using paper and pencil, I want to learn how one would go about computing the general solution in Mathematica.

To this end I wrote the following code:

  x \[Conditioned] y
  ,{x,y} \[Distributed] MultinormalDistribution[{Subscript[μ, x], Subscript[μ,y]}, {{Subscript[σ,x]^2,Subscript[σ,x]^2 }, {Subscript[σ,x]^2 ,Subscript[σ,x]^2+ Subscript[σ, y]^2}}]

Unfortunately, the only output I get it just what I wrote as the input. There is no solution. How can I compute the conditional expectations I want?

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    – rhermans
    Jul 30, 2018 at 16:15
  • $\begingroup$ On a side note, you should avoid using Subscript while defining symbols (variables). We know they look nice, but Subscript[x, 1] is not a symbol, but a composite expression where Subscript is an operator without built-in meaning. You expect to do $x_1=2$ but you are actually doing Set[Subscript[x, 1], 2] which is to assign a DownValues to the operator Subscript and not an OwnValues to an indexed x as you may intend. Read how to properly define indexed variables here $\endgroup$
    – rhermans
    Jul 30, 2018 at 16:16

1 Answer 1


First, following @rhermans advice about Subscripts's is imperative. Second, while writing $x | y$ is the mathematical way to write a conditioned statement, the function Expectation requires the term on the right of the modulus sign to be a "condition". So putting all of this together we have

Expectation[x \[Conditioned] y == y0, 
{x, y} \[Distributed] MultinormalDistribution[{μx, μy}, {{σx^2, σx^2}, {σx^2, σx^2 + σy^2}}]]

(y0 σx^2 + μx σx^2 - μy σx^2 + μx σy^2)/(σx^2 + σy^2)

Using FullSimplify on the result gives

μx + ((y0 - μy) σx^2)/(σx^2 + σy^2)

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