Under what conditions can a Poisson distribution approximate a binomial distribution
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Examples, solutions, videos, activities, and worksheets that are suitable for A Level Maths. Show What are the conditions for which a Poisson Distribution can be used as an approximation to the Binomial distribution? Poisson approximation to the Binomial Distribution Poisson Approximation to the Binomial Distribution (Example)
Poisson approximation to Binomial: S2 Edexcel January 2013 Q1
The Relationship Between the Binomial and Poisson Distributions
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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 $n \ge 100$ and $np \le 10$, the Poisson distribution (taking $\lambda = np$) can provide a very good approximation to the binomial distribution. This is particularly useful as calculating the combinations inherent in the probability formula associated with the binomial distribution can become difficult when $n$ is large. To better see the connection between these two distributions, consider the binomial probability of seeing $x$ successes in $n$ trials, with the aforementioned probability of success, $p$, as shown below. $$P(x)={}_nC_x p^x q^{n-x}$$Let us denote the expected value of the binomial distribution, $np$, by $\lambda$. Note, this means that $$p=\frac{\lambda}{n}$$and since $q=1-p$, Now, if we use this to rewrite $P(x)$ in terms of $\lambda$, $n$, and $x$, we obtain $$P(x) = {}_nC_x \left( \frac{\lambda}{n} \right)^x \left( 1-\frac{\lambda}{n} \right)^{n-x}$$Using the standard formula for the combinations of $n$ things taken $x$ at a time and some simple properties of exponents, we can further expand things to $$P(x) = \frac{n(n-1)(n-2) \cdots (n-x+1)}{x!} \cdot \frac{\lambda^x}{n^x} \left( 1 - \frac{\lambda}{n} \right)^{n-x}$$Notice that there are exactly $x$ factors in the numerator of the first fraction. Let us swap denominators between the first and second fractions, splitting the $n^x$ across all of the factors of the first fraction's numerator. $$P(x) = \frac{n}{n} \cdot \frac{n-1}{n} \cdots \frac{n-x+1}{n} \cdot \frac{\lambda^x}{x!}\left( 1 - \frac{\lambda}{n} \right)^{n-x}$$Finally, let us split the last factor into two pieces, noting (for those familiar with Calculus) that one has a limit of $e^{-\lambda}$. $$P(x) = \frac{n}{n} \cdot \frac{n-1}{n} \cdots \frac{n-x+1}{n} \cdot \frac{\lambda^x}{x!}\left( 1 - \frac{\lambda}{n} \right)^n \left( 1 - \frac{\lambda}{n} \right)^{-x}$$It should now be relatively easy to see that if we took the limit as $n$ approaches infinity, keeping $x$ and $\lambda$ fixed, the first $x$ fractions in this expression would tend towards 1, as would the last factor in the expression. The second to last factor, as was mentioned before, tends towards $e^{-\lambda}$, and the remaining factor stays unchanged as it does not depend on $n$. As such, $$\lim_{n \rightarrow \infty} P(x) = \frac{e^{-\lambda} \lambda^x}{x!}$$ Which is what we wished to show. When can we approximate binomial with Poisson?The result is very close to the result obtained above dpois(x = 1, lambda = 1) =0.3678794. The appropriate Poisson distribution is the one whose mean is the same as that of the binomial distribution; that is, λ=np, which in our example is λ=100×0.01=1.
Why does Poisson approximation to binomial?The short answer is that the Poisson approximation is faster and easier to compute and reason about, and among other things tells you approximately how big the exact answer is.
What are the 3 conditions for a Poisson distribution?Poisson Process Criteria
Events are independent of each other. The occurrence of one event does not affect the probability another event will occur. The average rate (events per time period) is constant. Two events cannot occur at the same time.
Under what conditions can the Poisson random variable be used to approximate a probability associated with the binomial random variable?In addition, the Poisson distribution can be obtained as an approximation of a binomial distribution when the number of trials n of the latter distribution is large, success probability p is small, and np is a finite number.
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