(11C) Poisson distribution

12182017, 09:38 AM
(This post was last modified: 12312017 02:35 PM by Gamo.)
Post: #1




(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:
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 

12192017, 08:15 PM
Post: #2




RE: (11C) Poisson distribution
(12182017 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(k_{1} ≤ k ≤ k_{2}). Code: LBL A Example for λ = 2,5: Code: Enter λ 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 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 Stirlingbased approximation, and since there is no iteration the result is returned immediately. Code: LBL E Example: Evaluate P(80) for λ=90. Code: 90 [A] 90,0000 Dieter 

12202017, 01:21 AM
Post: #3




RE: (11C) Poisson distribution
Thanks Dieter
Very nice program detail with indepth information. Gamo 

12232017, 06:42 PM
Post: #4




RE: (11C) Poisson distribution
Gamo
Check the HP25 post for Poisson Distribution from 1977. BEST! SlideRule 

12242017, 12:27 PM
(This post was last modified: 12242017 12:53 PM by Dieter.)
Post: #5




RE: (11C) Poisson distribution
(12232017 06:42 PM)SlideRule Wrote: Check the HP25 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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