Poisson distribution with mean 7.8 - find modal value

Question

The discrete random variable X has a poisson distribution with mean $x$. What is an expression for the modal value of X if $x$ = 7.8.

What is a modal value? Does it mean mode? I've never come across this type of question - please help!!!! Thank you

Answer 1

The usual notation for the mean of a Poisson distribution is $\lambda$ (small Greek letter corresponding to 'ell'). I will use that, instead of $x,$ which might be confusing.

It is important to distinguish between the mode (point at which the maximum value of a function is obtained) and modal value (value of the function at the mode).

The mode of $Pois(\lambda = 7.8)$ is 7. That is, if $X \sim Pois(7.8),$ then $P(X = 7)$ has is the greatest probability for any single point of the distribution. The modal value is $P(X = 7) = 0.1428021.$

In general, for noninteger $\lambda$, the mode is at the next integer below $\lambda.$

For integer $\lambda$ two values share the largest probability. For example, if $\lambda = 7,$ then $P(X = 6) = P(X = 7) = 0.1490028,$ and all other values of $X$ have smaller probabilities . Textbooks vary on terminology here: some say there is no mode (having defined the mode as to be necessarily unique) and others say there is a 'double mode' at the two values.

For integer $\lambda$ the 'double mode' is at $\lambda$ and the next smaller integer.

For a mathematical proof, let $$f(k) = P(X = k|\lambda) = \exp(-\lambda)\lambda^k/k! = e^{-\lambda}\,\lambda^k/k!,$$ for $k = 0, 1, 2, \dots.$

Then look at the ratio $f(k+1)/f(k)$ to see where it changes from being greater than 1 to less than 1. (It is not difficult because there is lots of cancellation in the ratio.) Many maximization questions in math are answered by calculus, but we are dealing with discrete $k$ here, so methods of calculus do not apply.

Wikipedia does give the mode of a Poisson distribution, but in terms of 'ceiling' and 'floor' notation that may not be familiar to you. That is good notation to know, and you might want to look up the definitions.