Expected value of the norm of a centered vector.
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Let $X$ be a random vector in $mathbb{R}^n$ such that $mathbb{E}[X]=0$. Is it true that $mathbb{E}[ |X|]=0$ ? (Euclidean norm).
probability probability-theory statistics random-variables
asked Jan 18 at 13:54
C.S.C.S.
235
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add a comment |
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Let $X$ be a random vector in $mathbb{R}^n$ such that $mathbb{E}[X]=0$. Is it true that $mathbb{E}[ |X|]=0$ ? (Euclidean norm).
probability probability-theory statistics random-variables
asked Jan 18 at 13:54
C.S.C.S.
235
$endgroup$
$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
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– Bey
Jan 18 at 14:28
add a comment |
$begingroup$
Let $X$ be a random vector in $mathbb{R}^n$ such that $mathbb{E}[X]=0$. Is it true that $mathbb{E}[ |X|]=0$ ? (Euclidean norm).
probability probability-theory statistics random-variables
asked Jan 18 at 13:54
C.S.C.S.
235
$endgroup$
Let $X$ be a random vector in $mathbb{R}^n$ such that $mathbb{E}[X]=0$. Is it true that $mathbb{E}[ |X|]=0$ ? (Euclidean norm).
probability probability-theory statistics random-variables
probability probability-theory statistics random-variables
asked Jan 18 at 13:54
C.S.C.S.
235
asked Jan 18 at 13:54
C.S.C.S.
235
asked Jan 18 at 13:54
C.S.C.S.
235
asked Jan 18 at 13:54
C.S.C.S.
235
asked Jan 18 at 13:54
C.S.C.S.
235
235
$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
$endgroup$
– Bey
Jan 18 at 14:28
add a comment |
$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
$endgroup$
– Bey
Jan 18 at 14:28
$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
$endgroup$
– Bey
Jan 18 at 14:28
$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
$endgroup$
– Bey
Jan 18 at 14:28
add a comment |
2 Answers
2
active
oldest
votes
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HINT
Not true because
$$left|vec{X}right| = 0 iff vec{X} = 0$$
In other words, for example, let the coordinates of $vec{X}$ be distributed symmetrically, e.g. $mathcal{U}(-1,1)$. Then what is the expected value?
But the Euclidean norm is
$$
sqrt{sum_{k=1}^n X_k^2}
$$
and $X_k^2 > 0$ a.s.
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
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add a comment |
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Maybe I can take in $mathbb{R}$ the random variable $X$ in ${-1,1}$ with probability 1/2 each. then $|X|=1$ always.
answered Jan 18 at 14:00
C.S.C.S.
235
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT
Not true because
$$left|vec{X}right| = 0 iff vec{X} = 0$$
In other words, for example, let the coordinates of $vec{X}$ be distributed symmetrically, e.g. $mathcal{U}(-1,1)$. Then what is the expected value?
But the Euclidean norm is
$$
sqrt{sum_{k=1}^n X_k^2}
$$
and $X_k^2 > 0$ a.s.
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
$endgroup$
add a comment |
$begingroup$
HINT
Not true because
$$left|vec{X}right| = 0 iff vec{X} = 0$$
In other words, for example, let the coordinates of $vec{X}$ be distributed symmetrically, e.g. $mathcal{U}(-1,1)$. Then what is the expected value?
But the Euclidean norm is
$$
sqrt{sum_{k=1}^n X_k^2}
$$
and $X_k^2 > 0$ a.s.
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
$endgroup$
add a comment |
$begingroup$
HINT
Not true because
$$left|vec{X}right| = 0 iff vec{X} = 0$$
In other words, for example, let the coordinates of $vec{X}$ be distributed symmetrically, e.g. $mathcal{U}(-1,1)$. Then what is the expected value?
But the Euclidean norm is
$$
sqrt{sum_{k=1}^n X_k^2}
$$
and $X_k^2 > 0$ a.s.
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
$endgroup$
HINT
Not true because
$$left|vec{X}right| = 0 iff vec{X} = 0$$
In other words, for example, let the coordinates of $vec{X}$ be distributed symmetrically, e.g. $mathcal{U}(-1,1)$. Then what is the expected value?
But the Euclidean norm is
$$
sqrt{sum_{k=1}^n X_k^2}
$$
and $X_k^2 > 0$ a.s.
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
answered Jan 18 at 13:59
gt6989bgt6989b
36k22557
36k22557
add a comment |
add a comment |
$begingroup$
Maybe I can take in $mathbb{R}$ the random variable $X$ in ${-1,1}$ with probability 1/2 each. then $|X|=1$ always.
answered Jan 18 at 14:00
C.S.C.S.
235
$endgroup$
add a comment |
$begingroup$
Maybe I can take in $mathbb{R}$ the random variable $X$ in ${-1,1}$ with probability 1/2 each. then $|X|=1$ always.
answered Jan 18 at 14:00
C.S.C.S.
235
$endgroup$
add a comment |
$begingroup$
Maybe I can take in $mathbb{R}$ the random variable $X$ in ${-1,1}$ with probability 1/2 each. then $|X|=1$ always.
answered Jan 18 at 14:00
C.S.C.S.
235
$endgroup$
Maybe I can take in $mathbb{R}$ the random variable $X$ in ${-1,1}$ with probability 1/2 each. then $|X|=1$ always.
answered Jan 18 at 14:00
C.S.C.S.
235
answered Jan 18 at 14:00
C.S.C.S.
235
answered Jan 18 at 14:00
C.S.C.S.
235
answered Jan 18 at 14:00
C.S.C.S.
235
235
add a comment |
add a comment |
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$begingroup$
Norms are always non-negative, taking the value 0 iff their input vector is zero: en.wikipedia.org/wiki/Norm_(mathematics)#Definition
$endgroup$
– Bey
Jan 18 at 14:28