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Chapter 3: Problem 11

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

Short Answer

Expert verified
The mode of the distribution is 1.

Step by step solution

01

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 \(\lambda^2 = 2\).
02

Solve for 位

To solve for \(\lambda\), take the square root of both sides of the equation \(\lambda^2 = 2\). This results in \(\lambda = \sqrt{2}\).
03

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.

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Most popular questions from this chapter

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