The Poisson Random Variable -A Poisson random variable takes nonnegative integer values. The PMF Px(k)=e-1 k! ,k=0,1,2,…, Normalization condition 路-(+++ ●Xa k=0 =e-e1=1
The Poisson Random Variable A Poisson random variable takes nonnegative integer values. The PMF 𝑝𝑋 𝑘 = 𝑒 −𝜆 𝜆 𝑘 𝑘! ，𝑘 = 0, 1, 2, … , Normalization condition 𝑘=0 ∞ 𝑒 −𝜆 𝜆 𝑘 𝑘! = 𝑒 −𝜆 1 + 𝜆 + 𝜆 2 2! + 𝜆 3 3! + ⋯ = 𝑒 −𝜆𝑒 𝜆 = 1
-Poisson random variable can be viewed as a binomial random variable with very small p and very large n. More precisely,the Poisson PMF with parameter A is a good approximation for a binomial PMF with parameters n and p where 1 =np,n is large and p is small See the wiki page for a proof
Poisson random variable can be viewed as a binomial random variable with very small 𝑝 and very large 𝑛. More precisely, the Poisson PMF with parameter 𝜆 is a good approximation for a binomial PMF with parameters 𝑛 and 𝑝 where 𝜆 = 𝑛𝑝, 𝑛 is large and 𝑝 is small. See the wiki page for a proof
Examples Because of the above connection,Poisson random variables are used in many scenarios. X is the number of typos in a book of n words. The probability that any one word is misspelled is very small. X is the number of cars involved in accidents in a city on a given day. The probability that any one car is involved in an accident is very small
Examples Because of the above connection, Poisson random variables are used in many scenarios. 𝑋 is the number of typos in a book of 𝑛 words. The probability that any one word is misspelled is very small. 𝑋 is the number of cars involved in accidents in a city on a given day. The probability that any one car is involved in an accident is very small
The Poisson Random Variable For Poisson random variable px(k)=e k! aλ≤1，monotonically decreasing A>1,first increases and then decreases Px(k) 4 Px(k) Poissonλ=0.5 Poissonλ=3 e-1~0.6 e-~0.05 l 0123 01234567k
The Poisson Random Variable For Poisson random variable 𝑝𝑋 𝑘 = 𝑒 −𝜆 𝜆 𝑘 𝑘! 𝜆 ≤ 1, monotonically decreasing 𝜆 > 1, first increases and then decreases
Content ■ Basic Concepts Probability Mass Function Functions of Random Variables Expectation,Mean,and Variance Joint PMFs of Multiple Random Variables ■ Conditioning Independence
Content Basic Concepts Probability Mass Function Functions of Random Variables Expectation, Mean, and Variance Joint PMFs of Multiple Random Variables Conditioning Independence