# Introduction to probability and distribution theory pdf

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Published: 01.12.2020  It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject.

## Probability theory

Sign in. In my first and second introductory posts I covered notation, fundamental laws of probability and axioms. These are the things that get mathematicians excited. However, probability theory is often useful in practice when we use probability distributions.

Probability distributions are used in many fields but rarely do we explain what they are. Often it is assumed that the reader already knows I assume this more than I should. For example, a random variable could be the outcome of the roll of a die or the flip of a coin.

A probability distribution is a list of all of the possible outcomes of a random variable along with their corresponding probability values. To give a concrete example, here is the probability distribution of a fair 6-sided die.

To be explicit, this is an example of a discrete univariate probability distribution with finite support. I can have an outcome of 1. It gets weird. You can probably guess when we get to continuous probability distributions this is no longer the case. In this case, we only have the outcome of the die roll.

In contrast, if we have more than one variable then we say that we have a multivariate distribution. The support is essentially the outcomes for which the probability distribution is defined. So the support in our example is. And since this is not an infinite number of values, it means that the support is finite. In the above example of rolling a six-sided die, there were only six possible outcomes so we could write down the entire probability distribution in a table.

In many scenarios, the number of outcomes can be much larger and hence a table would be tedious to write down. Worse still, the number of possible outcomes could be infinite, in which case, good luck writing a table for that.

To get around the problem of writing a table for every distribution, we can define a function instead. The function allows us to define a probability distribution succinctly. On a very abstract level, a function is a box that takes an input and returns an output.

For the vast majority of cases, the function actually has to do something with the input for the output to be useful. Graphically our function as a box looks like this:. Now it would be tedious to draw the digram above for every function that we want to create. So the above diagram can now be written as. This is better, however, we still have the problem that we have to draw a diagram to understand what the function is doing. We can define our function mathematically as.

One of the main takeaways from this is that with a function we can see how we would transform any input. For example, we could write a function in a programming language that takes a string of text as input and outputs the first letter of that string.

Here is an example of this function in the Python programming language. Given that one of the main benefits of functions is to allow us to know how to transform any input, we can also use this knowledge to visualise the function explicitly. Graphically it looks like this:. One of the most important features of functions are parameters. The reason that parameters are important is that they play a direct role in determining the output. This difference means that the outputs we get are completely different for the same input.

Parameters are arguably the most important feature of a probability distribution function because they define the output of the function which tells us the likelihood of certain outcomes in a random process. When we use a probability function to describe a discrete probability distribution we call it a probability mass function commonly abbreviated as pmf. Therefore, a probability mass function is written as:. I know this is getting a little horrible and mathematical but bear with me.

The probability mass function, f, just returns the probability of the outcome. Since a probability mass function returns probabilities it must obey the rules of probability the axioms that I described in my previous post.

Namely, the probability mass function outputs values between 0 and 1 inclusive and the sum of the probability mass function pmf over all outcomes is equal to 1. Mathematically we can write these two conditions as. We can also represent the die roll example graphically. Some probability distributions crop up so often that they have been extensively studied and have names.

One discrete distribution that crops up a lot is called the Bernoulli distribution. It describes the probability distribution of a process that has two possible outcomes.

An example of this is a coin toss where the outcome is heads or tails. The probability mass function of a Bernoulli distribution is. Here, x represents the outcome and takes the value 1 or 0. So in the case of a fair coin where the probability of landed heads or tails is 0. Often we want to be explicit about the parameters that are included in the probability mass function so we write.

Notice that we use the semicolon to separate the input variables from the parameters. Sometimes we are concerned with the probabilities of random variables that have continuous outcomes. Examples include the height of an adult picked at random from a population or the amount of time that a taxi driver has to wait before their next job. For these examples, the random variable is better described by a continuous probability distribution.

When we use a probability function to describe a continuous probability distribution we call it a probability density function commonly abbreviated as pdf. The normal distribution is probably the most common distribution in all of probability and statistics. One of the main reasons it crops up so much is due to the Central Limit Theorem. The probability density function for the normal distribution is defined as.

Where the parameters i. The normal distribution is an example of a continuous univariate probability distribution with infinite support. By infinite support, I mean that we can calculate values of the probability density function for all outcomes between minus infinity and positive infinity. The first thing to notice is that the numbers on the vertical axis start at zero and go up. This is a rule that a probability density function has to obey. Any output value from a probability density function is greater than or equal to zero.

In mathematical lingo we would say that the output is non-negative or write this mathematically as. However, unlike probability mass functions, the output of a probability density function is not a probability value. To get the probability from a probability density function we need to find the area under the curve. Mathematically we would write this as. Perhaps I need to write a brief series covering introductory calculus. Mathematically this is. Remember that we still have to follow the rules of probability distributions, namely the rule that says that the sum of all possible outcomes is equal to 1.

Therefore the following has to be true for the function to be a probability density function. This says that the area under the curve between minus infinity and positive infinity is equal to 1. An important thing to know about continuous probability distributions and something that may be really weird to come to terms with conceptually is that the probability of the random variable being equal to a specific outcome is 0.

For example, if we try to get the probability that the outcome is equal to the number 2 we would get. This may seem weird conceptually but if you understand calculus then it should make a little more sense.

Instead, what I want you to take away from this fact is that we can only talk about probabilities occurring between two values. Or we can ask about the probability of an outcome being greater than or less than a specific value. Explicitly I mean. So the probability of the random variable taking on a value between a and b exclusive is the same as the probability of it taking on a value between a and b inclusive.

That was much longer than I intended. Now that you have the basic understanding of what a probability distribution is, check out this great article by Sean Owen which covers the common probability distributions used in data science. For a more extensive list of probability distributions check out this Wikipedia page the list is quite long. As always, thanks for reading this far. Please feel free to leave comments, suggestions and questions.

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Check your inbox Medium sent you an email at to complete your subscription. Your home for data science. A Medium publication sharing concepts, ideas and codes. Get started. Open in app. ## Probability concepts explained: probability distributions (introduction part 3)

Publisher: American Mathematical Society. Comprehensiveness rating: 4 see less. The strength of this book in my view which is from an engineering perspective is that it approaches topics in a very natural way, using practical examples, simple graphics, and discussion of computer simulation when introducing topics. It does not seem to burden the reader with statistical jargon or needlessly deep discussions of theory, but it does not give the impression that it is trying to avoid these things either. In my opinion, the book omits all the right things, including most of the tables found in introductory probability and statistics texts.

The module is not intended to be a competitor to third-party libraries such as NumPy, SciPy, or proprietary full-featured statistics packages aimed at professional statisticians such as Minitab, SAS and Matlab. Quantitative Analysis. No attempt has been made to issue corrections for errors in typing, punctuation, etc. Mathematical statistics by Rietz, H. Lifetime hours 0— — — — — — — Frequency 97 64 51 14 14 In particular, I assume that the following concepts are familiar to the reader: distribution functions, convergence in probability, convergence in distribution, almost sure conver-. ## Probability concepts explained: probability distributions (introduction part 3)

Sign in. In my first and second introductory posts I covered notation, fundamental laws of probability and axioms. These are the things that get mathematicians excited. However, probability theory is often useful in practice when we use probability distributions.

The Probability component of MA consists of five parts, covering the following topics: Part 1: Introduction The need for probability; experiments, sample spaces, outcomes and events; Venn diagrams; relationships between sets; axioms of probability; relative frequency; subjective probability. Part 2: Sample spaces with no structure Deductions from the axioms; random sampling with and without replacement; conditional probability; pairwise and mutual independence; the law of total probability; Bayes' theorem. Part 3: Random variables Discrete and continuous random variables; the probability function; the cumulative distribution function; mean and variance; expectation; Bernouilli trials; geometric, binomial and Poisson distributions; the Poisson approximation to the binomial.

#### Module outline

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