Mixing
Let $M sim operatorname{Geometric}(p)$ and $ X vert M = m sim $ Uniform discrete on {1,m}
$begingroup$
Let $M sim operatorname{Geometric}(p)$ and $ X vert M = m sim $ Uniform discrete on {1,m}.
I want to find the density of X and then estimate the value of $p$ knowing one sample $X = x$.
If the density were continuous, I would do something like this:
$$ f_X = int_{text{over m}} f_M *f_{X mid M = m},dm $$
However the densities are discrete so instead of the integral I have a summation.
This is what I found:
$$ f_X(k) = sum_{m=1}^{infty} frac{1}{m} p(1-p)^{m-1}enspace (1<k<m)$$
First of all I am not so sure this is ok, so can someone check this result?
Then, how could I estimate the value of $p$ knowing the value of an output $x$?
probability density-function
edited Jan 26 at 15:46
Bernard
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
$endgroup$
add a comment |
$begingroup$
Let $M sim operatorname{Geometric}(p)$ and $ X vert M = m sim $ Uniform discrete on {1,m}.
I want to find the density of X and then estimate the value of $p$ knowing one sample $X = x$.
If the density were continuous, I would do something like this:
$$ f_X = int_{text{over m}} f_M *f_{X mid M = m},dm $$
However the densities are discrete so instead of the integral I have a summation.
This is what I found:
$$ f_X(k) = sum_{m=1}^{infty} frac{1}{m} p(1-p)^{m-1}enspace (1<k<m)$$
First of all I am not so sure this is ok, so can someone check this result?
Then, how could I estimate the value of $p$ knowing the value of an output $x$?
probability density-function
edited Jan 26 at 15:46
Bernard
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
$endgroup$
$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32
add a comment |
$begingroup$
Let $M sim operatorname{Geometric}(p)$ and $ X vert M = m sim $ Uniform discrete on {1,m}.
I want to find the density of X and then estimate the value of $p$ knowing one sample $X = x$.
If the density were continuous, I would do something like this:
$$ f_X = int_{text{over m}} f_M *f_{X mid M = m},dm $$
However the densities are discrete so instead of the integral I have a summation.
This is what I found:
$$ f_X(k) = sum_{m=1}^{infty} frac{1}{m} p(1-p)^{m-1}enspace (1<k<m)$$
First of all I am not so sure this is ok, so can someone check this result?
Then, how could I estimate the value of $p$ knowing the value of an output $x$?
probability density-function
edited Jan 26 at 15:46
Bernard
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
$endgroup$
Let $M sim operatorname{Geometric}(p)$ and $ X vert M = m sim $ Uniform discrete on {1,m}.
I want to find the density of X and then estimate the value of $p$ knowing one sample $X = x$.
If the density were continuous, I would do something like this:
$$ f_X = int_{text{over m}} f_M *f_{X mid M = m},dm $$
However the densities are discrete so instead of the integral I have a summation.
This is what I found:
$$ f_X(k) = sum_{m=1}^{infty} frac{1}{m} p(1-p)^{m-1}enspace (1<k<m)$$
First of all I am not so sure this is ok, so can someone check this result?
Then, how could I estimate the value of $p$ knowing the value of an output $x$?
probability density-function
probability density-function
edited Jan 26 at 15:46
Bernard
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
edited Jan 26 at 15:46
Bernard
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
edited Jan 26 at 15:46
Bernard
123k741117
edited Jan 26 at 15:46
Bernard
123k741117
edited Jan 26 at 15:46
Bernard
123k741117
123k741117
asked Jan 26 at 15:40
qcc101qcc101
629213
asked Jan 26 at 15:40
qcc101qcc101
629213
asked Jan 26 at 15:40
qcc101qcc101
629213
629213
$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32
add a comment |
$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32
$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32
add a comment |
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$begingroup$
You would need to have a prior distribution for $p$ to be able to get the distribution of $p$ given an output $x$ (if you had a prior you could use Bayes' Theorem to get the conditional distribution of $p$).
$endgroup$
– Alex
Jan 26 at 16:00
$begingroup$
Why density? Assuming mass points of $M$ start at $1$, the probability mass function is $P(X=k)=sumlimits_{m=k}^infty P(X=kmid M=m)P(M=m)$ for $k=1,2,ldots$ by total probability.
$endgroup$
– StubbornAtom
Jan 26 at 16:27
$begingroup$
Why m starts equal to k? How would you write explicitly the density?
$endgroup$
– qcc101
Jan 27 at 6:32