Approximately transforming discrete random integers to discrete random reals
$begingroup$
Consider the random positive integer $n$ which is binomially distributed:
$$P(n) = {N choose n} p^n q^{N-n}, $$
and define an approximate transformation to a real random variable (arbitrary), for example, as
$$ x approx pi n.$$
Note this transformation being approximate means $x/pi$ is not definitely an integer, although it is probably close to an integer.
Blindly applying the law for transformations of random variables says
$$P(x) approx sum_{n=0}^N P(n)delta_{n,x/pi} = {Nchoose x/pi} p^{x/pi}q^{N-x/pi}.$$
Suddenly I'm evaluating factorials of non-integer numbers.
This seems fine enough if they are interpreted using Gamma functions, such as
$$ P(x) approx frac{Gamma(1+N)}{Gamma(1+ N - x/pi)Gamma(1+x/pi)}p^{x/pi}q^{N-x/pi},$$
but a priori there is no reason to do this. Is there any way to understand this transformation in an even slightly more rigorous way? I'm thrown off by an arbitrary swap of factorials to gamma functions, and by the evaluation of a kronecker delta on a real number: I'm used to visualizing this as an identity matrix.
probability statistics probability-distributions
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
$endgroup$
add a comment |
$begingroup$
Consider the random positive integer $n$ which is binomially distributed:
$$P(n) = {N choose n} p^n q^{N-n}, $$
and define an approximate transformation to a real random variable (arbitrary), for example, as
$$ x approx pi n.$$
Note this transformation being approximate means $x/pi$ is not definitely an integer, although it is probably close to an integer.
Blindly applying the law for transformations of random variables says
$$P(x) approx sum_{n=0}^N P(n)delta_{n,x/pi} = {Nchoose x/pi} p^{x/pi}q^{N-x/pi}.$$
Suddenly I'm evaluating factorials of non-integer numbers.
This seems fine enough if they are interpreted using Gamma functions, such as
$$ P(x) approx frac{Gamma(1+N)}{Gamma(1+ N - x/pi)Gamma(1+x/pi)}p^{x/pi}q^{N-x/pi},$$
but a priori there is no reason to do this. Is there any way to understand this transformation in an even slightly more rigorous way? I'm thrown off by an arbitrary swap of factorials to gamma functions, and by the evaluation of a kronecker delta on a real number: I'm used to visualizing this as an identity matrix.
probability statistics probability-distributions
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
$endgroup$
$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49
add a comment |
$begingroup$
Consider the random positive integer $n$ which is binomially distributed:
$$P(n) = {N choose n} p^n q^{N-n}, $$
and define an approximate transformation to a real random variable (arbitrary), for example, as
$$ x approx pi n.$$
Note this transformation being approximate means $x/pi$ is not definitely an integer, although it is probably close to an integer.
Blindly applying the law for transformations of random variables says
$$P(x) approx sum_{n=0}^N P(n)delta_{n,x/pi} = {Nchoose x/pi} p^{x/pi}q^{N-x/pi}.$$
Suddenly I'm evaluating factorials of non-integer numbers.
This seems fine enough if they are interpreted using Gamma functions, such as
$$ P(x) approx frac{Gamma(1+N)}{Gamma(1+ N - x/pi)Gamma(1+x/pi)}p^{x/pi}q^{N-x/pi},$$
but a priori there is no reason to do this. Is there any way to understand this transformation in an even slightly more rigorous way? I'm thrown off by an arbitrary swap of factorials to gamma functions, and by the evaluation of a kronecker delta on a real number: I'm used to visualizing this as an identity matrix.
probability statistics probability-distributions
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
$endgroup$
Consider the random positive integer $n$ which is binomially distributed:
$$P(n) = {N choose n} p^n q^{N-n}, $$
and define an approximate transformation to a real random variable (arbitrary), for example, as
$$ x approx pi n.$$
Note this transformation being approximate means $x/pi$ is not definitely an integer, although it is probably close to an integer.
Blindly applying the law for transformations of random variables says
$$P(x) approx sum_{n=0}^N P(n)delta_{n,x/pi} = {Nchoose x/pi} p^{x/pi}q^{N-x/pi}.$$
Suddenly I'm evaluating factorials of non-integer numbers.
This seems fine enough if they are interpreted using Gamma functions, such as
$$ P(x) approx frac{Gamma(1+N)}{Gamma(1+ N - x/pi)Gamma(1+x/pi)}p^{x/pi}q^{N-x/pi},$$
but a priori there is no reason to do this. Is there any way to understand this transformation in an even slightly more rigorous way? I'm thrown off by an arbitrary swap of factorials to gamma functions, and by the evaluation of a kronecker delta on a real number: I'm used to visualizing this as an identity matrix.
probability statistics probability-distributions
probability statistics probability-distributions
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
edited Dec 20 '18 at 5:05
kevinkayaks
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
asked Dec 20 '18 at 3:52
kevinkayakskevinkayaks
1558
1558
$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49
add a comment |
$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49
$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49
$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49
add a comment |
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$begingroup$
I am not sure if what you are looking for is a continuous real distribution that resembles a given discrete distribution. One way to accomplish this is by using kernel density estimation - see en.wikipedia.org/wiki/Kernel_density_estimation
$endgroup$
– mlerma54
Dec 20 '18 at 4:49