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 Poisson distribution is \(\sqrt{6} - 1\).

Step-by-step solution

Recap Poisson distribution

A variable \(X\) follows a Poisson distribution with parameter \(\lambda\), the mean number of events in an interval, can be written as: \(P(X=x) = \frac {e^{-\lambda}\lambda^x} {x!}\). This formula is used to calculate the probability of getting \(x\) number of outcomes.

Given condition

It is given in the problem that \(P(X=1) = P(X=3)\), substituting these in the Poisson distribution formula gives: \(\frac {e^{-\lambda}\lambda^1} {1!} = \frac {e^{-\lambda}\lambda^3} {3!}\), simplify the equation before solving.

Solve for \(\lambda\)

Simplify the equation by crossing out common terms, which gives \(\lambda^2 = 6\). Solving this equation for \(\lambda\) yields two solutions, as \(\lambda\) can't be negative in Poisson Distribution, \(\lambda = \sqrt{6}\).

Find the mode

Once we have the value of \(\lambda\), we can find the mode. For a Poisson distribution, the mode is the largest integer less than or equal to \(\lambda\). Therefore, the mode in this case is \(\lambda -1 = \sqrt{6} - 1\).