# Under what conditions do the binomial and Poisson distribution give approximately the same results?

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## Under what conditions do the binomial and Poisson distribution give approximately the same results?

The Poisson distribution is actually a limiting case of a Binomial distribution when the number of trials, n, gets very large and p, the probability of success, is small. As a rule of thumb, if n≥100 and np≤10, the Poisson distribution (taking λ=np) can provide a very good approximation to the binomial distribution.

The Poisson distribution is a limiting case of the binomial distribution which arises when the number of trials n increases indefinitely whilst the product μ = np, which is the expected value of the number of successes from the trials, remains constant.

**Under which two conditions will the binomial probability be approximately equal to the Poisson probability?**

When the number n of trials is very large and the probability p small, e.g. n > 25 and p < 0.1, binomial probabilities are often approximated by the Poisson distribution.

**What are the conditions for using the Poisson distribution?**

Conditions for Poisson Distribution: Events occur independently. In other words, if an event occurs, it does not affect the probability of another event occurring in the same time period. The rate of occurrence is constant; that is, the rate does not change based on time.

## Why the normal distribution can be used to approximate the binomial or Poisson distributions?

Because for certain discrete distributions, namely the Binomial and Poisson distributions, summing large values can be tedious or not practical. Thankfully, the Normal Distribution allows us to approximate the probability of random variables that would otherwise be too difficult to calculate.

### When would you use a Poisson distribution?

If your question has an average probability of an event happening per unit (i.e. per unit of time, cycle, event) and you want to find probability of a certain number of events happening in a period of time (or number of events), then use the Poisson Distribution.

**How do you approximate Poisson distribution?**

The Poisson(λ) Distribution can be approximated with Normal when λ is large. For sufficiently large values of λ, (say λ>1,000), the Normal(μ = λ,σ2 = λ) Distribution is an excellent approximation to the Poisson(λ) Distribution.

**How do you approximate the binomial distribution?**

Then the binomial can be approximated by the normal distribution with mean μ=np and standard deviation σ=√npq. Remember that q=1−p. In order to get the best approximation, add 0.5 to x or subtract 0.5 from x (use x+0.5 or x−0.5).