In this post, I’ll be talking about two of the more common parametric distributions, the Binomial and Poisson distributions. We kind of covered the Normal distribution last post so I’ll skip that for now unless I come across some world-changing info about it.
So, what are Parametric Distributions? Usually, probabilities depend on some unknown constant. For example, the probability of flipping a Heads depends on if the coin is fair or two-headed. The unknown constant in this example is called the parameter. Depending on the type of problem, you can have a different parameter, and the distributions I will mention today are the most common (probably).
The Binomial Distribution
In order for an experiment to be considered a Binomial, it has to meet a few requirements:
- The outcome must be either a success or failure
- Each trial must be identical
- The probability of success is the same for each trial
- Each trial must be independent
The Binomial Distribution is really prevalent in the real world. The most common example I’ve seen is a series of coin flips, aka “What’s the probability of getting 3 heads in 5 flips of a fair coin?”
To answer this question, we need to know three things. The probability of success, the probability of failure, and the number of ways I can get 3 heads in 5 tries. The probability of success is just 1/2 because the coin is fair, and because of this, the probability of failure is also 1/2. Now, I can write out all the different ways to get 3 heads from 5 flips, but I won’t – and I’ll use a shortcut if you remember Combinations. The number of ways to get 3 heads out of 5 flips is 5C3 = 5!/(5-3)!3! = 10. You can verify this if you want.
The Binomial Distribution is really helpful because theres an equation that gives the probability of k successes within N trials with a probability of success (parameter) of θ. Here it is:
P(X=k|θ) = (N choose k) * θ^k * (1-θ)^(N-k)
Which just means the probability of k successes in N tries with an underlying chance of success of θ = (the number of ways you get get k groupings from N) * (the probability of k successes) * (the probability of N-k failures). The last part is because in order to have 3 successes out of 5, you need to also have 2 failures.
One thing I want to mention here is there’s a special type of Binomial Distribution called the Bernoulli Distribution which is when N=1. More to come on that later but it’s worth a shout out.
The Poisson Distribution
As the Binomial Distribution deals with counting successes over a certain amount of trials, the Poisson Distribution helps to understand successes over a period of time or space. The parameter we use in the Poisson is referred to as λ, which represents the rate. This is good for estimating things like how many customers show up to a counter over an hour.
There’s a way to show it, but the Poisson equation is derived directly from the Binomial distribution. I’m not going to try to explain it. Instead, here’s the equation:
P(X=k|λ) = λ^k * e^-λ/k!
From what I’ve gathered, this equation is used to estimate λ. You can choose a bunch of different values for λ and plot the resulting probability to find out what values of λ are the most likely for the given k. If given a λ, it should be relatively straightforward to calculate a probability, but I’m not sure how someone in the real world would get to a λ. It seems kind of circular to me but that’s probably just because I still haven’t nailed down the applications of this distribution.
Regardless, the idea of trying to estimate λ given k kind of touches on Likelihood estimates which I don’t fully understand yet but it seems like a Poisson problem can get much more complicated. I guess for now, it’s just good to have a little foundation on what this is.
That’s it for now.