”If we want to encode the vector [6,2,8,4] in a single expression we can create a function containing those numbers: f(x) = 6 + 2x² + 8x³ + 4x⁴
…or if you flip the vector and use x=10:
6284Probability-generating functions
21–30 of 54 posts
Re: Probability-generating functions
#22I think the 12th order G(t) example is missing another term with coefficient 1/5 -- since these coefficients must sum to 1
Unless it’s already been fixed, the queen’s term (t^12) has 2/5 in the article.
Re: Probability-generating functions
#23For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
but isn't a characteristic function just "the" way to bridge the gap between sets, functions, and logic(? ...a 3way bridge!?) I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S) how the heck is that a fourier transform!??
You're thinking of a "characteristic function" in the sense of "indicator function" of a subset (https://en.wikipedia.org/wiki/Indicator_function), which is different thing to the characteristic function of a probability density function.
Re: Probability-generating functions
#24For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
Yes, this provides good intuition about why it is useful: the PDF of the sum of two random variables is the convolution of the original PDFs. A convolution is awkward to work with, but by the convolution theorem it is a multiplication in the Fourier domain. This immediately suggests that the Fourier transform of a PDF would be a useful thing to work with.
If you don't say that this is what you are doing then it all seems quite mysterious.
Re: Probability-generating functions
#25”If we want to encode the vector [6,2,8,4] in a single expression we can create a function containing those numbers: f(x) = 6 + 2x² + 8x³ + 4x⁴ …or if you flip the vector and use x=10: 6284
Indeed. However, note that this is limited to encoding values between 0 and 9.
Re: Probability-generating functions
#26Probably worth mentioning the moment-generating function as well, since it's a bit more elementary than the characteristic function and shares many of the properties. It's also more simply related to the probability generating function, you can go from one to the other with a basic change of coordinates (t -> log(x)). I also estimate calculating the moment generating function to be easier in most cases. In fact most…
One can also think of probability generating functions as (flipped) Z transforms, moment generating functions as (flipped Laplace transforms), and characteristic functions as Fourier transforms of the respective PMF/PDF. Lot of their properties then follow from simple properties of Signals and Systems.
Do you have a reference that explains this in more detail? I'd be curious to know.
Re: Probability-generating functions
#27For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
but isn't a characteristic function just "the" way to bridge the gap between sets, functions, and logic(? ...a 3way bridge!?) I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S) how the heck is that a fourier transform!??
You can think of it like this:
- The characteristic function of a random variable X is defined as the function that maps t --> ExpectedValue[ exp( i * t * X ) ]
- Computing this expected value is the same as regarding t as a constant and integrating the function x --> exp( i * t * x) with respect to the distribution of X, i.e. if X has the density f, we compute the integral of f(x) * exp( i * t * x) with respect to x over the domain of f.
- on the other hand: computing the Fourier transform of f (here representing the density of X) and evaluating it at point t (i.e. computing (F(f))(t) if F represents the Fourier transform) is the same as fixing t and computing the integral of f(x) * exp( -i * t * x) with respect to x.
- Rearranging the integrand in the previous expression to f(x) * exp( i * -t * x), we see that it is the same as the integrand used in the characteristic function, only with a -t instead of a t.
Hope that helps :)
Re: Probability-generating functions
#28For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
As a physicist, the moment when everything just clicked was when I realised that connected Feynman diagrams were basically the cumulants of that distribution.
Then almost everything in physics is about "what is the characteristic/moment/cumulant generating function?" and associated Legendre transforms
Re: Probability-generating functions
#29Lost the plot at "... and for sensible v, this is equivalent to ..." :(
This is just the sum of a geometric sequence.
Re: Probability-generating functions
#30For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
but isn't a characteristic function just "the" way to bridge the gap between sets, functions, and logic(? ...a 3way bridge!?) I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S) how the heck is that a fourier transform!??
“Characterstic function” is (was) an overloaded term.
What you described is more often referred to as an “indicator function” these days, with “characteristic functions” denoting the transform (Fourier, laplace, z - depending on context). Closely related to “moment generating functions” to the point of being almost interchangeable.