# Identify Poisson or Exponential Distribution and determine lambda

I am trying to identify the distribution of my variable, $$X$$. It measures goals per minute of soccer players. Possible values are $$[0,inf]$$ and they are non integers. I believe this to be an exponential distribution because Poisson measures integers and we are dealing with the successive events over continual time not discrete.

If this is a Poisson distribution, what should I use as the data when plotting or comparing distributions? Every time I use the rate metric, I trigger an error for things like poisson.test() or fitdist(x,"pois") in R. When I use exponential distributions, everything works.

Also, is there a way to estimate $$\lambda$$ generally or do you do it for discrete time periods?

• This looks like a probability density, which would not be in units of "goals per minute." – whuber Jan 25 at 22:21
• Well if the statistic measures the number of goals a player scored divided by the number of cumulative minutes they played, it is referred to as goals per minute. When analyzed on a per minute basis, we get a distribution. Do we not? – Jack Armstrong Jan 25 at 22:28
• $f$ would then be in units of probability per (goals per minute). It is, however, not a plausible candidate to describe that, in part because it assigns appreciable probability to cumulative playing times that are impossible (they extend beyond the length of any conceivable game). For both those reasons something seems wrong about your description. – whuber Jan 26 at 14:07
• Generally to calculate goals per minute we sum a players goals and divide by their total minutes played in a season. So 10 goals/450 minutes = 0.022. – Jack Armstrong Jan 26 at 14:09

Because your variable is a count that you are expressing as a rate, the Poisson distribution would be a natural choice. If $$Y$$ represents the number of goals scored by player $$i$$, you could model $$Y_i \sim \mbox{Poisson}(N_i\lambda)$$, where $$N_i$$ is the total number of minutes that this player was on the field, and $$\lambda$$ is the rate parameter that can be interpreted as goals per minute.
• Is the $\lambda$ parameter only defined at an interval? If I were to empirical test, I could use poisson.test() in R, with x being my data, but what would be my Time, T, value? – Jack Armstrong Jan 25 at 22:50
• You can think of this $\lambda$ as representing the true goals per minute rate of the players. What kind of test are you trying to perform? Are you trying to carry out a hypothesis test of some kind, or is your goal to estimate the $\lambda$ parameter? – Izzy Jan 25 at 23:10
• You are correct that only a count can arise from a Poisson distribution. As @whuber mentioned too, a Poisson regression might be the right direction. Your variable -- number of goals -- is a count. The true rate would be represented by $\lambda$. If $Y$ is the number of goals and $N$ is the number of minutes, then $Y \sim \mbox{Poisson}(N\lambda)$ is the natural way to model your data. Note that $Y$ is not the rate itself; the number of minutes is being accounted for through $N$. In Poisson regression, we would think of the minutes $N$ as the offset. – Izzy Jan 26 at 17:27
• Here is a simple example in R:  goals <- c(3,4,5,2,1,3) minutes <- c(10,20,30,5,5,15) glm(goals~.,data=data.frame(goals=goals),family=poisson,offset=log(minutes))  If you exponentiate the intercept term from the resulting fit, that represents your estimate $\lambda$ of the true rate of goals per minute. Then for any individual player, their number of goals could be modeled as arising from a Poisson distribution with parameter $N \lambda$, where $\lambda$ is this estimated value and $N$ is the minutes they play. – Izzy Jan 26 at 17:29
• Your approach to plotting should work. It's just that your current plot will not look very meaningful since the rate of goals per minute $\lambda$ is so small that it's very unlikely to have more than 0 goals per minute. You could make your plot easier to interpret by picking some amount of time, such as 100 minutes or whatever makes sense in your context, and recreating the same plot but with dpois(x,100*lambda). Then this will show you the distribution of number of goals per 100 minutes. – Izzy Jan 27 at 17:37