+- HP Forums (https://www.hpmuseum.org/forum)
+-- Forum: HP Software Libraries (/forum-10.html)
+--- Forum: General Software Library (/forum-13.html)
+--- Thread: (11C) Poisson distribution (/thread-9720.html)
(11C) Poisson distribution - Gamo - 12-18-2017 09:38 AM
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!
/
RTNRun 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
RE: (11C) Poisson distribution - Dieter - 12-19-2017 08:15 PM
(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 1Example 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
/
RTNExample:
Evaluate P(80) for λ=90.
Code:
90 [A] 90,0000
80 [E] 0,0250 P(80)Dieter
RE: (11C) Poisson distribution - Gamo - 12-20-2017 01:21 AM
Thanks Dieter
Very nice program detail with in-depth information.
Gamo
RE: (11C) Poisson distribution - SlideRule - 12-23-2017 06:42 PM
Gamo
Check the HP-25 post for Poisson Distribution from 1977.
BEST!
SlideRule
RE: (11C) Poisson distribution - Dieter - 12-24-2017 12:27 PM
(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