Deriving the Poisson distribution, can I symbolically compute this limit?

[Kyle Andrews]

Part of a derivation of the Poisson distribution involves computing the limit of:

p := product((n-i)/n, i=1..(x-1))

I can numerically "confirm" this limit by playing around with the values of n and x:

numeric eval(p, [n, x], [1000000, 30])

But I can't seem to come to a symbolic result:

limit(p, n=%plusInfinity) -- hint: should be 1

Is there some machinery already in FriCAS that would make this work?

[oldk1331]

It should be solvable:

p1 := eval(p, n = 1/n)
limit(p1, n = 0)

The result is "1^(x-1)"

I think this is a bug. There should be something added for 'exprToGenUPS',
like r1645 for 'exprToUPS'.

[Kyle Andrews]

Thanks for looking. I don't want to be to hard on FriCAS since I know this involves 2 variables. I know I can do better than numeric eval:

If I eval(p, x=30) :: FRAC POLY INT and then take the limit it works, and I can map across many x's to check.

But it if course would be nice to check every possible x in one go.

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[Kyle Andrews]

FYI, here was the code I was thinking about in my last reply.

map(j +-> limit(eval(p, x=j) :: FRAC(POLY(INT)), n = %plusInfinity), [-5, 0, 1, 15, 20, 23, 30, 50, 100])

[oldk1331]

> I don't want to be to hard on FriCAS since I know this involves 2 variables.

It's not about 2 variables.

p := product((n-i)/n, i=1..(x-1))
limit(p, n=%plusInfinity)

That is equvalent to

p1 := product(1-i*n, i=1..(x-1)) -- eval(p, n = 1/n)
limit(p1, n=0)

FriCAS can handle that limit.
But due to the defect I mentioned, it can handle limit
from both side of 0, but not from one side of 0:

limit(p1, n=0, "right") -- gives "failed"

[Waldek Hebisch]

I am not sure which one is worse: not getting one sided limit
or how we get two sided one. Namely, 'exprToUPS' contains
"last chance" expander which expands function into series
by taking successive derivatives. This expander should be
used for computing limits only when we know that there is
an expansion (and when there is an expansion we must be careful
as this expander can fail due to division by 0). Gruntz-Shackell
routines (mrv_*) perform checks that function is of expected form.
'limit' blindly passes expression to expander...

--
Waldek Hebisch