Introduction XI – Distributions and what they model

A lot of random variables follow a known distribution. It is therefore good to know some of the most common distributions and what they model. Remembering their PMFs or PDFs is not necessary.

Uniform Distribution

The uniform distribution is probably one of the most famous distributions. It models a random variable with equally likely outcomes. A great example is a die. If the die is fair then the probability for an outcome is the same as the probability for any other outcome.

Example: Suppose we have a fair die. What is the probability for each outcome?

Answer: The probability for each outcome is \frac{ 1 }{ 6 } and the random variable is therefore uniform distributed.

Bernoulli Distribution

A random variable with just two outcomes, success and failure, is bernoulli distributed. The bernoulli distribution models one trail and the probability of success and failure in this one trial.

Example: Suppose we toss a coin. Let X be that the coin shows heads. Since a coin flip has just two outcomes and we suppose that heads is a success, X is bernoulli distributed.

Binomial Distribution

The binomial distribution is related to the bernoulli distributions. It models the probability of k successes in n independent bernoulli trials.

Example: Suppose we toss a coin. Let X be the number of heads in 100 tosses. X is binomial distributed because it is basically a bernoulli distribution but with 100 trials. k would be the number of heads.

The Binomial Distribution looks as following:

The first one has 100 trials and the second one 30 trials.

The binomial distribution has the following PMF where (\begin{matrix} n \\ k \end{matrix}) is the number of ways to get k successes in n trials: P(X=k)=(\begin{matrix} n \\ k \end{matrix}) p^{ k }(1-p)^{ n-k }

Geometric Distribution

The geometric distribution models the numbers of failures (k) before the first success.

Example: Suppose we toss a fair coin. What is the probability that we have 5 tails before our first heads?

The geometric distribution has the following PMF where (1-p) is the probability of failure: P(X=k)=(1-p)^{ k }p

geometric

Normal (Gaussian) Distribution

The normal or gaussian distribution is probably the most important distribution and is used in many scientific fields.

Example: The error of a measurement is often supposed to be normal distributed, as well as the distance of an outcome to the mean of the random variable.

normal

The graphic show the standard normal distribution with \mu=0 and \sigma^{ 2 }=1

There are loads of more distributions out there, like the poisson distribution and the hyper geometric distribution. A look at wikipedia often helps to find their properties and PMFs or PDFs as well as much more helpful information.

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