Property 1: Suppose the probability p of success of a single trial approaches 0 while the number of trials n approaches infinity and the value μ = np stays fixed. Then the binomial distribution B(n, p) approaches the Poisson distribution with mean μ.
Proof: Let f(x) be the pdf for a binomial distribution as described in the statement of the theorem. Then
Now
where z = -p. But
and so
Also
Thus as p → 0 and n → 
which completes the proof.
References
Hoel, P. (1962) Introduction to mathematical statistics, 3rd Ed. John Wiley and Sons
Chamberlain, A. (2016) Deriving the Poisson distribution from the binomial distribution
https://medium.com/@andrew.chamberlain/deriving-the-poisson-distribution-from-the-binomial-distribution-840cc1668239
ProofWiki (2021) Binomial distribution approximated by Poisson distribution
https://proofwiki.org/wiki/Binomial_Distribution_Approximated_by_Poisson_Distribution#:~:text=Binomial%20Distribution%20Approximated%20by%20Poisson%20Distribution,-From%20ProofWiki&text=Let%20X%20be%20a%20discrete,%CE%BBkk!e%E2%88%92%CE%BB
Dear George.
I have built a poissons distribution model in excel to try n guide me on my betting on soccer matches.Am not yet very good at it.Pse try n use the data below for a match and cal the propable outcome of the game:
Home Draw Away Home or Draw Double chance Away or Draw Under 2.5 over 2.5 BTTS BTTN ODD EVEN
1.23 5.83 16 1.04 1.11 1.9 1.9 1.9 2.76 1.34 1.92 1.76
NB: ODD/EVEN: are prob that the out come of the game after the 90 mins will be an even or odd number.I would love to follow your method if the results are close to the outcome.
Regards.
Kenneth,
Sorry, but I don’t understand the data nor the goal. When you say that the outcome is odd or even, are you referring to the total points scored by both sides?
Charles
good work. i would like to get these lecture notes for statistical analysis through my email.
thank you.
George,
Sorry, but I can’t make the lecture notes available via email. I plan to provide a book which contains these lecture notes which you can purchase for a modest fee, but the book is not yet available.
Charles