# Probability and Statistics

## Introduction

Probability and statistics are very important within many branches of science and engineering. Probability is also used to create powerful models. One real life example would be global warming forecasts. Instead of providing that global warming “probably” will have a considerable effect, it would with the use of statistical models be possible to say that “there is a 95% chance that the sea level will rise with a certain amount of centimeters, and that the average temperature will increase by a certain amount of degrees.

One example of an application area is within electrical engineering, where it is used within electronics and communication. Statistics is also extensively used within insurance and finance to calculate prices. The price of a stock for example, can be described as a random process, with a certain mean and variance changing over time. Within insurance it is used to calculate the risks and determine the appropriate price for different insurance services.

## Basics

### What are random variables?

A random variable (or random number) is different from a regular number. A regular number always has a fixed value, however a random variable can take any kind of value, and which kind of values it can take is described by its so called distribution. Hence, every-time we draw a random variable, the result can take on a new value. The random variable can be seen as a unknown lottery ticket number before it is read drawn from a box, and the value is assigned first when the ticket number is read. Before it is read it is unknown, since we do not know which number is on the ticket. However, we might know something about the ticket numbers in the box, for example in which range of values the ticket numbers are (1 to 100 for example).

### How are random variables defined?

There are many types of random variables. The overall characteristics of a random variable is usually defined by a distribution. A distribution describes how the outcome of billions of realizations of the variable would come out to be. A given distribution is usually defined by its type, e.g. Gaussian, Poisson and uniform, as well as a few parameters such as mean, variance and decay rate. These characteristics are described by mathematical formulas, hence a basic understanding of calculus is helpful to enable full understanding of probability.

The area of probability might seem confusing in the beginning since there is a lot of mathematics that is used to describe it. However, it is important not to be afraid of the mathematics, but at the same time it is not good to spend to much time to dwelve to deep into it either. For most people the important thing is to develop a feeling for what is possible and not possible with the help of statistics and probability, the details can be learned at a later stage.

When discussing about random variables, there are two main types, continuous and discrete, it is important to differentiate between these. Continuous simply means that for each time, t, there exists a value for the variable. However for the discrete case, the variable is only defined as a series, e.g

X[n] , n = 1, 2, 3..

Where for the continuous case we have

X(t), t = [0,T]

Both continuous and discrete random variables are important and have their different uses. Two important ways to describe continuous random variables are probability density functions and cumulative distribution functions.

#### Probability density function

The probability density function (pdf) of which if you integrate it within an interval, say [a,b]. It will give you the probability that the variable is within the interval. Hence, if you are given a probability density function you can calculate the chance that the variable is within any interval.

#### Cumulative distribution function

The cumulative distribution function (cdf) is closely connected to the probability density function. It is defined as follows

F(x) = P( X < x)

Where P(X < x) denotes the probability that the random variable X is less than the constant x. Hence, it is possible to define the cumulative distribution function as the integral of the probability density function from minus infinity to the constant x.