(11C) Poisson distribution
12-18-2017, 09:38 AM (This post was last modified: 12-31-2017 02:35 PM by Gamo.)
Post: #1
 Gamo Senior Member Posts: 324 Joined: Dec 2016
(11C) Poisson distribution
The Poisson distribution is popular for modelling the number of times an event occurs in an interval of time or space.

Formula: P(k events in interval) = (e^-λ)(λ^k) / k!

where:
λ (lambda) is the average number of events per interval
e is the number 2.71828... (Euler's number) the base of the natural logarithms
k takes values 0, 1, 2, …
k! = k × (k − 1) × (k − 2) × … × 2 × 1 is the factorial of k.

Example Problem:
Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5
Because the average event rate is 2.5 goals per match, λ = 2.5
What is the probability of gold of P(k) = 0, 1, 2, 3, 4, 5, 6, 7

Program:
Code:
 LBL A (λ) STO 1 RTN LBL B (k) STO 2 RTN LBL C (P) RCL 1 CHS e^x RCL 1 RCL 2 Y^x x RCL 2 X! / RTN

Run Program:
2.5 A
0 B
C 0.082
1 B
C 0.205
2 B
C 0.257
3 B
C 0.213
.
.
.
.
7 B
C 0.010

The table below gives the probability for 0 to 7 goals in a match.

k P(k goals in a World Cup soccer match)
0 0.082
1 0.205
2 0.257
3 0.213
4 0.133
5 0.067
6 0.028
7 0.010

Credit to Wikipedia for information and example problem.

Gamo
12-19-2017, 08:15 PM
Post: #2
 Dieter Senior Member Posts: 2,174 Joined: Dec 2013
RE: (11C) Poisson distribution
(12-18-2017 09:38 AM)Gamo Wrote:  What is the chance of gold of P(k) = 0, 1, 2, 3, 4, 5, 6, 7

Gold ?-)  I assume this is supposed to mean
"What is the probability P(k) for k = 0, 1, 2, 3, 4, 5, 6 or 7 goals".

But why do you use two separate labels for k and P(k)? This way calculating the PDF always requires pressing two keys, B and C.

Here is another version that also calculates the CDF, i.e. P(k1 ≤ k ≤ k2).

Code:
LBL A STO 0 RTN LBL B RCL 0 x<>y y^x LastX x! / RCL 0 CHS e^x * RTN LBL C ABS INT EEX 3 / x<>y ABS INT + STO I LastX GSB B x=0? RTN ENTER ENTER LBL 1 ISG      // (ISG I on the 15C) GTO 2 x<>y RTN LBL 2 RCL 0 * RCL I INT / + LastX GTO 1

Example for λ = 2,5:

Code:
Enter λ  2,5        [A]    2,5000 Calculate the probability for 1, 2 or 3 goals   1         [B]    0,2052   P(1)   2         [B]    0,2565   P(2)   3         [B]    0,2138   P(3) Calculate the probability of a match with 1 to 4 goals 1 [ENTER] 4 [C]    0,8091   P(1 ≤ k ≤ 4) If desired: [R↓]   0,1336   P(4)             [R↓]   0,2052   P(1)

Direct evaluation of the Poisson PDF often leads to overflow errors. Even cases where k>69 can not be handled this way. Too bad there is no lnΓ function available, this could provide an easy fix.

But there are two workarounds:

1. The recursive method of the CDF routine significantly extends the useable range for λ and k, and this can also be used for calculating the PDF:
Simply enter 0 [ENTER] k [C], and when the result is displayed press [R↓] or [x<>y] to get P(k).

Example:
Evaluate P(80) for λ=90.

Code:
   90        [A]  90,0000    80        [B]   8,1940 -40  flashing                    Overflow error while trying to evaluate 90^80 and 80! 0 [ENTER] 80 [C]   0,1582             [x<>y] 0,0250   P(80)

However, the iteration with k required loops may take some time on a hardware 11C.

Also please note that for λ > 227,9559242 the expression e–λ will underflow to zero. In this case 0 is returned.

2. For large λ and k the following code may be used. It implements a Stirling-based approximation, and since there is no iteration the result is returned immediately.

Code:
LBL E STO I RCL 0 RCL I / LN 1 + * RCL 0 - e^x RCL I 6 1/x + Pi * 2 * sqrt / RTN

Example:
Evaluate P(80) for λ=90.

Code:
   90        [A]  90,0000    80        [E]   0,0250   P(80)

Dieter
12-20-2017, 01:21 AM
Post: #3
 Gamo Senior Member Posts: 324 Joined: Dec 2016
RE: (11C) Poisson distribution
Thanks Dieter

Very nice program detail with in-depth information.

Gamo
12-23-2017, 06:42 PM
Post: #4
 SlideRule Senior Member Posts: 403 Joined: Dec 2013
RE: (11C) Poisson distribution
Gamo

Check the HP-25 post for Poisson Distribution from 1977.

BEST!
SlideRule
12-24-2017, 12:27 PM (This post was last modified: 12-24-2017 12:53 PM by Dieter.)
Post: #5
 Dieter Senior Member Posts: 2,174 Joined: Dec 2013
RE: (11C) Poisson distribution
(12-23-2017 06:42 PM)SlideRule Wrote:  Check the HP-25 post for Poisson Distribution from 1977.

That's three short programs in this thread.
I just posted an improved version. Which also includes a correct initialization of the summation registers to avoid erroneous results.
And there's even some information on the author of the original programs.

Dieter
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