Mixing
Can I adjust a weighted average of uniforms to be uniform?
I have uniformly-distributed random variables $x_i$ satisfying
$$0le x_ile 1$$
I take a weighted mean of these using weights $w_i$ satisfying
$$0le w_ile 1$$
$$sum w_i=1$$
The naive weighted mean is
$$hat{x}=sum w_ix_i$$
but $hat{x}$ no longer has a uniform distribution.
What smooth adjustment $X=f(hat{x})$ can I make such that $X$ has uniform distribution?
probability-theory probability-distributions
asked Nov 20 '18 at 14:37
spraff
465213
add a comment |
I have uniformly-distributed random variables $x_i$ satisfying
$$0le x_ile 1$$
I take a weighted mean of these using weights $w_i$ satisfying
$$0le w_ile 1$$
$$sum w_i=1$$
The naive weighted mean is
$$hat{x}=sum w_ix_i$$
but $hat{x}$ no longer has a uniform distribution.
What smooth adjustment $X=f(hat{x})$ can I make such that $X$ has uniform distribution?
probability-theory probability-distributions
asked Nov 20 '18 at 14:37
spraff
465213
If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27
add a comment |
I have uniformly-distributed random variables $x_i$ satisfying
$$0le x_ile 1$$
I take a weighted mean of these using weights $w_i$ satisfying
$$0le w_ile 1$$
$$sum w_i=1$$
The naive weighted mean is
$$hat{x}=sum w_ix_i$$
but $hat{x}$ no longer has a uniform distribution.
What smooth adjustment $X=f(hat{x})$ can I make such that $X$ has uniform distribution?
probability-theory probability-distributions
asked Nov 20 '18 at 14:37
spraff
465213
I have uniformly-distributed random variables $x_i$ satisfying
$$0le x_ile 1$$
I take a weighted mean of these using weights $w_i$ satisfying
$$0le w_ile 1$$
$$sum w_i=1$$
The naive weighted mean is
$$hat{x}=sum w_ix_i$$
but $hat{x}$ no longer has a uniform distribution.
What smooth adjustment $X=f(hat{x})$ can I make such that $X$ has uniform distribution?
probability-theory probability-distributions
probability-theory probability-distributions
asked Nov 20 '18 at 14:37
spraff
465213
asked Nov 20 '18 at 14:37
spraff
465213
asked Nov 20 '18 at 14:37
spraff
465213
asked Nov 20 '18 at 14:37
spraff
465213
asked Nov 20 '18 at 14:37
spraff
465213
465213
If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27
add a comment |
If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27
If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27
add a comment |
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If $F$ is the cumulative distribution function of $hat x$, then $F(hat x)$ is uniform on $[0,1]$. See probability integral transform.
– Rahul
Nov 20 '18 at 15:01
So what is the CDF of $hat{x}$?
– spraff
Nov 20 '18 at 22:27