Poisson Distribution Derivation from Binomial Distribution

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Welcome to our exploration of how the Poisson distribution emerges from the binomial distribution. The binomial distribution describes the number of successes in n independent trials, each with probability p. When we let n approach infinity while p approaches zero, but keep their product np constant at lambda, something remarkable happens - the binomial distribution transforms into the Poisson distribution. This limiting process reveals one of the most elegant connections in probability theory. Here we see the binomial distribution with different parameters. Notice how as n increases from 10 to 50 to 100, while p decreases proportionally to keep np equal to 3, the binomial distribution approaches the Poisson distribution shown in black. The convergence becomes more apparent as n gets larger. This demonstrates the fundamental limit theorem that connects these two important probability distributions. Here's the mathematical derivation. We start with the binomial probability mass function. Substituting p equals lambda over n, we get the second expression. Expanding the binomial coefficient and rearranging terms gives us the third line. As n approaches infinity, the ratio of consecutive integers approaches 1, and we can use the fundamental limit that 1 minus lambda over n to the power n approaches e to the minus lambda. This gives us the Poisson probability mass function.