Statistics 101 — Binomial and Poisson Distribution

One can get the Poisson from Binomial by taking the limit, and the Binomial from Poisson by conditioning. More precisely, we have the following.

• If 𝑋∼Pois(𝜆1), 𝑌∼Pois(𝜆2)are independent random variables, then the distribution of 𝑋X given 𝑋+𝑌=n is 𝑋cond∼Bin(𝑛,𝜆1/(𝜆1+𝜆2))
• If X∼Bin(n,p), and if 𝑛→∞, 𝑝→0, such that 𝑛𝑝→𝜆, then (𝑋=𝑘)→e^−𝜆*(𝜆^𝑘/𝑘!)

How to define the distribution is Binomial or Poisson?

Binomial: a Bernoulli trial is repeated n times, there is a constant probability of success p, X is defined as the total number of successful of trial.

Poisson: Event occurs at random but with a constant average 𝜆 per some unit. X is defined as the number of events that occur per unit.

PDF or CDF

1)PDF( probability density function)
This basically is a probability law for a continuous random variable ( for discrete, it is probability mass function).

1. The probability law defines the chances of the random variable taking a particular value k.
   This definition is not valid for continuous random variables because the probability at a given point is zero.

2) CDF ( Cumulative Distribution Function)

This is simply the probability up to a particular value of the random variable. Generally denoted by F, F= P (X<=x) for any value of x in the X space. It is defined for both discrete and continuous random variables.