Probability Densities and Distributions Of A Random Variable Lecture 6/19
Introduction Until now, probability functions or probability measures were created by applying the probability axioms on the set of elementary outcomes that make up a random experiment's sample space. For example, if two fair dice were tossed we assigned to each of the 36 possible outcomes a probability value of 1/36. Then we further applied the axioms to assign probability measures (probabilities) to events comprised of these elementary outcomes; the event of the that the sum of the dice is 7 has a probability of... In doing this we created the probability space for our model random experiment. In this lecture we will learn about a fundamental concept that is used to redefine the sample space of an experiment. Then we will investigate the consequences of redefining a sample space, resulting in changes in the probability structure. To illustrate this, in many games played with dice, we are interested in the sum of the dice and not the individual points shown on each die. If this situation we are interested in calculating the possible probabilities for the sum of the points on the roll of two fair dice. We can use our previous sample space S, (the list of all the possible outcomes of rolling two dice) to calculate the probabilities for this "new sample space".  Let X be a "random" variable that represents the sum of the two dice. For each event as described by the value of X, we calculate the following probabilities:
X = n
In the table above, the values taken on by X, partition the sample space into a collection of mutually exclusive events.  The frequency is a count of the number of outcomes in S that correspond to the event X = n, for n = 2, 3, ... , 12. The random variable X, the sum of the two dice, generates a new sample space (outcomes = the sum) with a new probability structure. In general, rules for redefining sample spaces are called random variables. Formally, Definition A real-valued function X:S ® R whose domain is the sample space S is called a random variable, if, for each interval IÍ R, { s | X(s)Î I } is an event.
This definition applies to both discrete and continuous sample spaces and we refer to randoms variables as discrete or continuous accordingly.
As a conceptual framework, random variables are of fundamental importance, they are used as the building blocks of the theory of probability.
Our efforts from this point on is to introduce the important definitions, concepts and techniques assocoated with random variables. Altogether, these ideas comprise the mathematical foundation of probability and statistics.
Let's look at some more examples involving random variables.
Associated with each random variable are two types of functions:
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