# Analytical Methods in Probability Theory: Proceedings by Dogeu D. (ed.), Lucaks E. (ed.), Rohatgi V.K. (ed.) By Dogeu D. (ed.), Lucaks E. (ed.), Rohatgi V.K. (ed.)

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Additional info for Analytical Methods in Probability Theory: Proceedings

Example text

In such a case, the random variable X is said to be a discrete random variable. In the first part of this book, we are mainly concerned with discrete random variables that take on a finite number of values. Let us assume that X can only take on values from the finite set I = {x1 , . . , xM }. The event X = xj is defined as the set of those outcomes for which the random variable X takes on the value xj . The probability of the event X = xj is thus defined as the sum of the probabilities of the individual outcomes for which X takes on the value xj .

Solution. Let the random variable X denote the largest of the two scores. This random variable has I = {1, . . , 6} as its set of possible values. To find the distribution of X, you will need the sample space of the experiment. A logical choice is the set S = {(1, 1), . . , (1, 6), (2, 1), . . , (6, 1), . . , (6, 6)}, where the outcome (i, j ) corresponds with the event that the score of John is i dots and the score of Mary is j dots. Each of the 36 possible outcomes is equally probable with fair dice.

That is, an event is a set consisting of possible outcomes of the experiment. If the outcome of the experiment is contained in the set E, it is said that the event E has occurred. A sample space in conjunction with a probability measure is called a probability space. A probability measure is simply a function P that assigns a numerical probability to each subset of the sample space. A probability measure must satisfy a number of consistency rules that will be discussed later. 2 Basic probability concepts 29 Let’s first illustrate a few things in light of an experiment that children sometimes use in their games to select one child out of the group.