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1. Basics of probability and statistics

The purpose of this and the following chapter is to briefly go through the most basic concepts in probability theory and statistics that are important for you to understand. If these concepts are new to you, you should make sure that you have an intuitive feeling of their meaning before you move on to the following chapters in this book.

1.1. Random variables and probability distributions

The first important concept of statistics is that of a random experiment. It is referred to as any process of measurement that has more than one outcome and for which there is uncertainty about the result of the experiment. That is, the outcome of the experiment can not be predicted with certainty. Picking a card from a deck of cards, tossing a coin, or throwing a die, are all examples of basic random experiments.

The set of all possible outcomes of an experiment is called the sample space of the experiment. In case of tossing a coin, the sample space would consist of a head and a tail. If the experiment was to pick a card from a deck of cards, the sample space would be all the different cards in a particular deck. Each outcome of the sample space is called a sample point.

An event is a collection of outcomes that resulted from a repeated experiment under the same condition. Two events would be mutually exclusive if the occurrence of one event precludes the occurrence of the other event at the same time. Alternatively, two events that have no outcomes in common are mutually exclusive. For example, if you were to roll a pair of dice, the event of rolling a 6 and of rolling a double have the outcome (3,3) in common. These two events are therefore not mutually exclusive.

Events are said to be collectively exhaustive if they exhaust all possible outcomes of an experiment. For example, when rolling a die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes. Hence, the set of all possible die rolls is both mutually exclusive and collectively exhaustive. The outcomes 1 and 3 are mutually exclusive but not collectively exhaustive, and the outcomes even and not-6 are collectively exhaustive but not mutually exclusive.

Even though the outcomes of any random experiment can be described verbally, such as described above, it would be much easier if the results of all experiments could be described numerically. For that purpose we introduce the concept of a random variable. A random variable is a function that assigns unique numerical values to all possible outcomes of a random experiment.

By convention, random variables are denoted by capital letters, such as X, Y, Z, etc., and the values taken by the random variables are denoted by the corresponding small letters x, y, z, etc. A random variable from an experiment can either be discrete or continuous. A random variable is discrete if it can assume only a finite number of numerical values. That is, the result in a test with 10 questions can be 0, 1, 2, ..., 10. In this case the discrete random variable would represent the test result. Other examples could be the number of household members, or the number of sold copy machines a given day. Whenever we talk about random variables expressed in units we have a discrete random variable. However, when the number of unites can be very large, the distinction between a discrete and a continuous variable become vague, and it can be unclear whether it is discrete or continuous.

A random variable is said to be continuous when it can assume any value within an interval. In theory that would imply an infinite number of values. But in practice that does not work out. Time is a variable that can be measured in very small units and go on for a very long time and is therefore a continuous variable. Variables related to time, such as age is therefore also considered to be a continuous variable. Economic variables such as GDP, money supply or government spending are measured in units of the local currency, so in some sense one could see them as discrete random variables. However, the values are usually very large so counting each Euro or dollar would serve no purpose. It is therefore more convenient to assume that these measures can take any real number, which therefore makes them continuous.

Since the value of a random variable is unknown until the experiment has taken place, a probability of its occurrence can be attached to it. In order to measure a probability for a given events, the following formula may be used:

This formula is valid if an experiment can result in n mutually exclusive and equally likely outcomes, and if m of these outcomes are favorable to event A. Hence, the corresponding probability is calculated as the ratio of the two measures: n/m as stated in the formula. This formula follows the classical definition of a probability.

Example 1.1

You would like to know the probability of receiving a 6 when you throw a die. The sample space for a die is {1, 2, 3, 4, 5, 6}, so the total number of possible outcome are 6. You are interested in one of them, namely 6. Hence the corresponding probability equals 1/6.

Example 1.2

You would like to know the probability of receiving 7 when rolling two dice. First we have to find the total number of unique outcomes using two dice. By forming all possible combinations of pairs we have (1,1), (1,2),..., (5,6),(6,6), which sum to 36 unique outcomes. How many of them sum to 7? We have (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): which sums to 6 combinations. Hence, the corresponding probability would therefore be 6/36 = 1/6.

The classical definition requires that the sample space is finite and that each outcome in the sample space is equally likely to appear. Those requirements are sometimes difficult to stand up to. We therefore need a more flexible definition that handles those cases. Such a definition is the so called relative frequency definition of probability or the empirical definition. Formally, if in n trials, m of them are favorable to the event A, then P(A) is the ratio m/n as n goes to infinity or in practice we say that it has to be sufficiently large.

Example 1.3

Let us say that we would like to know the probability to receive 7 when rolling two dice, but we do not know if our two dice are fair. That is, we do not know if the outcome for each die is equally likely. We could then perform an experiment where we throw two dice repeatedly, and calculate the relative frequency. In Table 1.1 we report the results for the sum from 2 to 7 for different number of trials.

Relative frequencies for different number of trials

Table 1.1 Relative frequencies for different number of trials

From Table 1.1 we receive a picture of how many trials we need to be able to say that that the number of trials is sufficiently large. For this particular experiment 1 million trials would be sufficient to receive a correct measure to the third decimal point. It seem like our two dices are fair since the corresponding probabilities converges to those represented by a fair die.

 
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