Poisson distribution. Does it have probability density function? Why or why not? Discuss your answer in the context of mean and variance of the Poisson distribution.
The function f(x) is called the probability density function (p.d.f) provided it satisfies the following two conditions
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- F(x) ≥ 0 -------------------b
- If the range of the continuous random variable is (a,b) ∫ f(x) = 1 ---------------a
A random variable X is said to have a binomial distribution and is referred as a binomial random variable, if and only if it’s probability distribution is given by
B (x;n,p) = nCx . px . (1-p)n-x for x = 0,1,2,…..n
The probability of getting ‘x’ success and ‘n-x’ failures in n trials is given by px . (1-p)n-x.
The number of trials in the case of binomial distribution is very large and cumbersome. Then the probability distribution used to approximate binomial possibilities. This limiting form of the binomial distribution when n → ∞ and p → 0, while n.p remains constant. Let λ = n.p, therefore p = λ/n. The binomial distribution written as
B(x,n,p) = nCx (λ/n)x (1-( λ/n))n-x = n (n-1) (n-2) (n-3)………(n-x+1)/x! x (λ/n) x (1-( λ/n))n-x
= 1.(1-1/n)(1-2/n)(1-3/n)……(1-(x-1)/n/x! x (λ) x x [(1 - ( λ/n))n/λ] -λ x (1-(λ/n))-x .
When n →∞ while x and p are constants
(1-1/n) (1-2/n) (1-3/n) ….(1-(x-1)/n) → 1
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