Example of Uniform

A uniform distribution (often called 'rectangular') is one in which all values between two boundaries occur roughly equally. For example, if you roll a six-sided die, you're equally likely to get 1, 2, 3, 4, 5, or 6. If you rolled it 6,000 times, you'd probably get roughly 1,000 of each result. The results would form a uniform distribution from 1 to 6.

Another example of something that's uniformly distributed is the digits of pi. Each digit makes up about 10% of the values. To show that, I put together a quick visualization of the first 500 digits. In this, each digit should occur roughly 50 times:

Example of Normal

A normal distribution looks like a bell. It's often called a 'bell curve' or a 'Gaussian'. Many things in nature have nearly-normal distributions...heights of men in the US...measurement errors...IQs. A cool thing related to them though is the Central Limit Theorem. It roughly states that the means of many non-normal distributions are normally distributed.

As a simple example of that, I generated 20 random values between 0 and 9 (uniform distribution with a mean of 4.5) 1000 times. Each iteration, I took the mean of those 20 random values, and made a histogram of the means found so far. You can see that it is roughly normal (bell-shaped):

A U distribution is one in which points are more likely to be at the edges of a range than in the middle. For example, if 40% of students in a class get A's, 40% get zero, and the remaining 20% get something in between, that would form a U distribution.

A cool example of this distribution type is the position of an object with sinusoidal motion. Imagine measuring the angle of a pendulum every 1/100 seconds. It slows down on the sides, and speeds up in the middle, so more measurements will be at the edges than in the middle. I animated a perfect one here (a circle is sinusoidal motion in two dimensions...a pendulum is one...two dimensions looked cooler):

Example of Poisson

Poisson distributions give the probability of something occurring a certain number of times if it typically occurs at a fixed rate and each occurrence is independent of previous occurrences. An example use case is an online tutoring service that typically gets 4 students in the period between 9 pm and 9:30 pm and wants to calculate the probability of getting 6 students in that period.

Scores in the group stage of the World Cup can be modeled reasonably well with a Poisson distribution. I took the game scores from the last 6 World Cups and animated it below: