7
You can use ParameterMixtureDistribution
Mean @ ParameterMixtureDistribution[PoissonDistribution[λ],
Distributed[ λ, NormalDistribution[m, 1]]]
Undefined
We need to use a distribution with positive support for the distribution of λ:
Mean @ ParameterMixtureDistribution[PoissonDistribution[λ],
Distributed[ λ, LogNormalDistribution[m, 1]]]
E^(1/2 + m)
...
5
pd = EmpiricalDistribution[({1, 1, 1, 1, 1, 5}/10) -> Range[6]];
Probability[Total[Array[x, 4]] == 18, Thread[Array[x, 4] \[Distributed] pd]]
(* result 12/125 *)
4
I hope this does the trick. It's more code than yours but I've come at it from a slightly different angle - I suppose another implementation can't hurt right? I've used FindPermutation to get $K_n$ and SolveAlways for non-square $G_n$:
vec[W_] := Join @@ Transpose[W]
vech[W_] := With[{n = Length[W]},
Flatten[MapThread[#1[[-#2 ;;]] &, {Transpose[W], ...
4
This is just an extended comment: Because the question is about speeding things up, I think you'll need to consider merging different functions depending on what values of $v$ are of most interest. Here is an example of the timing:
f[r_, f_, v_] := Block[{t}, Coefficient[Expand[Sum[t^j/j!, {j, 0, v}]^(f - 1)] (f - 1)^(-(r - v))*
(r - v)!*Binomial[r, v]/f^...
3
Without filtering:
Total[Times @@@ Map[If[# == 6, 1/2, 1/10] &,
Flatten[Permutations /@
IntegerPartitions[18, {4}, Range[6]], 1], {2}]]
12/125
answered Sep 20 at 14:57
3
list = {P1, P2, P3, P4, P5};
Total[Flatten@
Table[(-1)^(k - 1) Times @@@ Subsets[list, {k}], {k, 1, 5}]]
Update-1 Thanks to the suggestions in comments.
list = {A1, A2, A3, A4, A5};
Sum[Total[(-1)^(k - 1) P /@ And @@@ Subsets[list, {k}]], {k, 1, 5}]
Update-2
need to be updated later :)
3
This is now closer to an answer in that I attempted to follow @UlrichNeumann 's good suggestion about splitting the integration into parts.
I changed the subscripted variables to x, y, and z to lighten the text load. The constraints in the Boole function can be written as
1 > x > y > z > 0 && z > 1 - x - y - z && x - z < 2 ...
2
Identities allow computing these matrices 100s of times faster
$$
\begin{eqnarray}
P_d&=&\text{nsymm[d], the symmetrizer matrix}\\
\mu&=&E[x]\\
X^2&=&E[xx']\\
E[(xx')(xx')]&=&2X ^2+X^2 \text{Tr} X^2 -2\|\mu\|^2 \mu \mu'\\
E[xx'\otimes xx']&=&2P_d( X^2\otimes X^2) + \text{vec} X^2 (\text{vec} X^2)'-2(\mu \mu')\otimes (\...
2
There are many works to start from:
Propagation risk of COVID-19 by local contact in Spain
100 Days of COVID19 Over US Counties
Maps for Visualizing Covid-19's Effect
Also see works of Hiroki Sayama:
CODE: https://github.com/hsayama/COVID-19-geographical-animations
VISUALS:
https://twitter.com/HirokiSayama/status/1261623727772631047
https://twitter.com/...
2
There's no need to use patterns here. They just make more trouble figuring them out and debugging. There's also no reason to introduce a coding scheme for the cards at this stage, which is a bit of a premature optimization.
ranks[hand_] := Sort[hand[[All, 1]]]
suits[hand_] := Sort[hand[[All, 2]]]
countranks[hand_] := Values@Counts@ranks@hand
pairQ[hand_] := ...
2
You can speed up the calculations for your initial equation several orders of magnitude (with ever larger increases in speed for larger values of k) by using Sort and Accumulate:
(* Generate a random sample of positive numbers *)
k = 100;
SeedRandom[12345];
x = RandomVariate[ChiSquareDistribution[20], k];
(* Original equation *)
t1 = AbsoluteTiming[Abs[1 - (...
1
PExpand[expr_] := Module[{A, B, expr1, f, H}, f = expr /. H_[___] -> H;
expr1 = expr /. f[A___] -> A;
f /@ Expand[expr1 //. A_ || B_ :> A (-B) + A + B] /.
f[-B__] :> -f[Times[B]]]
PExpand[P[Or @@ Array[Subscript[A, #1] &, 5]]]
1
What do you mean? The beta prime distribution is right here. Even if it wasn't, you could easily define it with TransformedDistribution as
betaPrimeDistribution[α_, β_] :=
TransformedDistribution[
\[FormalX]/(1 - \[FormalX]),
\[FormalX] \[Distributed] BetaDistribution[α, β]
]
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