Category Archives: statistics

Conditioning and a sum of Poisson random variables

Previously we talked about the Poisson distribution. The Poisson distribution with mean $\mu \gt 0$ is a distribution on the natural numbers whose density function is $$f(n) = \frac{e^{-\mu}\mu^n}{n!}$$ We have already seen that the Poisson distribution essentially arises from the binomial distribution as a sort of "limiting case". In fact, the Poisson distribution is […]

Maximum likelihood, moments, and the uniform distribution

Suppose we have observations from a known probability distribution whose parameters are unknown. How should we estimate the parameters from our observations? Throughout we'll focus on a concrete example. Suppose we observe a random variable drawn from the uniform distribution on $[0,\theta]$, but we don't know what $\theta$ is. Our one observation is the number […]

Where does the Poisson distribution come from?

The Poisson distribution is a discrete probability distribution on the natural numbers $0,1,2,\dots$. Its density function depends on one parameter $\mu$ and is given by $$d(n) = \frac{e^{-\mu}\mu^n}{n!}$$ Not surprisingly, the parameter $\mu$ is the mean, which follows from the exponential series $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ Here is what the density function looks like when […]

Do The Continents Affect Surface Air Temperature?

The internet has enabled researchers and organisations of various kinds to make their data available for free to download and hence anyone with a computer and some rudimentary R knowledge can observe and analyse all sorts of trends in everything from economics to society to natural phenomena. Obviously this can provide endless hours of fun […]