Expectation of normally distributed r.v conditioned on vector subspace
$begingroup$
Let $X$ be a random variable with normal distribution $N(mu, Sigma), mu in mathbb{R}^n$ and $V subset mathbb{R}^n$ be a vector subspace.
I want to calculate $mathrm{E}[X|X in V]$ and my intuition tells me that it should be equal to expectation of orthogonal projection of $X$ to $V$, but how can i prove that rigorously?
My attemp:
Let $pi$ be a projection on $V$ than $mathrm{E}[X = pi X + (1-pi) X |X in V] = mathrm{E}[pi X|X in V] = ???$.
May be that object does't exists because of $P(X in V) = 0$?
normal-distribution conditional-expectation
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
$endgroup$
add a comment |
$begingroup$
Let $X$ be a random variable with normal distribution $N(mu, Sigma), mu in mathbb{R}^n$ and $V subset mathbb{R}^n$ be a vector subspace.
I want to calculate $mathrm{E}[X|X in V]$ and my intuition tells me that it should be equal to expectation of orthogonal projection of $X$ to $V$, but how can i prove that rigorously?
My attemp:
Let $pi$ be a projection on $V$ than $mathrm{E}[X = pi X + (1-pi) X |X in V] = mathrm{E}[pi X|X in V] = ???$.
May be that object does't exists because of $P(X in V) = 0$?
normal-distribution conditional-expectation
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
$endgroup$
add a comment |
$begingroup$
Let $X$ be a random variable with normal distribution $N(mu, Sigma), mu in mathbb{R}^n$ and $V subset mathbb{R}^n$ be a vector subspace.
I want to calculate $mathrm{E}[X|X in V]$ and my intuition tells me that it should be equal to expectation of orthogonal projection of $X$ to $V$, but how can i prove that rigorously?
My attemp:
Let $pi$ be a projection on $V$ than $mathrm{E}[X = pi X + (1-pi) X |X in V] = mathrm{E}[pi X|X in V] = ???$.
May be that object does't exists because of $P(X in V) = 0$?
normal-distribution conditional-expectation
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
$endgroup$
Let $X$ be a random variable with normal distribution $N(mu, Sigma), mu in mathbb{R}^n$ and $V subset mathbb{R}^n$ be a vector subspace.
I want to calculate $mathrm{E}[X|X in V]$ and my intuition tells me that it should be equal to expectation of orthogonal projection of $X$ to $V$, but how can i prove that rigorously?
My attemp:
Let $pi$ be a projection on $V$ than $mathrm{E}[X = pi X + (1-pi) X |X in V] = mathrm{E}[pi X|X in V] = ???$.
May be that object does't exists because of $P(X in V) = 0$?
normal-distribution conditional-expectation
normal-distribution conditional-expectation
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
edited Dec 12 '18 at 18:47
qwenty
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
asked Dec 12 '18 at 18:10
qwentyqwenty
440417
440417
add a comment |
add a comment |
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