Question

Let \(X\) have a Poisson distribution. If \(P(X=1)=P(X=3)\), find the mode of the distribution.

Short answer

The mode of the distribution is 1.

Step-by-step solution

Use the given condition to set up an equation

It is given that \(P(X=1)=P(X=3)\). In the Poisson distribution, these probabilities can be expressed using the formula \(\lambda^ke^{-\lambda} / k!\). Thus the equation becomes \(\frac{\lambda^1e^{-\lambda}}{1!} = \frac{\lambda^3e^{-\lambda}}{3!}\). Simplifying, we find that \(\lambda^2 = 2\).

Solve for λ

To solve for \(\lambda\), take the square root of both sides of the equation \(\lambda^2 = 2\). This results in \(\lambda = \sqrt{2}\).

Find the mode of the distribution

For a Poisson distribution with parameter \(\lambda\), the mode is typically \(\lambda\), or \(\lambda - 1\) if \(\lambda\) is not an integer. So, here the mode would be \(\sqrt{2}\). However, by convention, when thinking about frequencies, we would round to the nearest whole number, yielding 1 as the mode in this particular instance.