Probability. Robert P. Dobrow

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СКАЧАТЬ frequency viewpoint will still be useful in order to gain intuitive understanding. And by the end of the book, we will actually derive the relative frequency viewpoint as a consequence of the mathematical theory.

1.3 PROBABILITY FUNCTION

      We assume for the next several chapters that the sample space is discrete. This means that the sample space is either finite or countably infinite.

      A set is countably infinite if the elements of the set can be arranged as a sequence. The natural numbers is the classic example of a countably infinite set. And all countably infinite sets can be put in one-to-one correspondence with the natural numbers.

      If the sample space is finite, it can be written as . If the sample space is countably infinite, it can be written as .

      The set of all real numbers is an infinite set that is not countably infinite. It is called uncountable. An interval of real numbers, such as (0,1), the numbers between 0 and 1, is also uncountable. Probability on uncountable spaces will require differential and integral calculus and will be discussed in the second half of this book.

      A probability function assigns numbers between 0 and 1 to events according to three defining properties.

      PROBABILITY FUNCTION

      Given a random experiment with discrete sample space , a probability function is a function on with the following properties:

      1 

      2 (1.1)

      3 For all events ,(1.2)

You may not be familiar with some of the notation in this definition. The symbol means “is an element of.” So means is an element of . We are also using a generalized -notation in Equations 1.1 and 1.2, writing a condition under the to specify the summation. The notation means that the sum is over all that are elements of the sample space, , that is, all outcomes in the sample space.

      In the case of a finite sample space , Equation 1.1 becomes

      And in the case of a countably infinite sample space , this gives

      In simple language, probabilities sum to 1. The third defining property of a probability function says that the probability of an event is the sum of the probabilities of all the outcomes contained in that event. We might describe a probability function with a table, function, graph, or qualitative description. Multiple representations are possible, as shown in the next example.

       Example 1.5 A type of candy comes in red, yellow, orange, green, and purple colors. Choose a piece of candy at random. What color is it? The sample space is Assuming the candy colors are equally likely outcomes, here are three equivalent ways of describing the probability function:

      1 0.200.200.200.200.20

      2 

      3 The five colors are equally likely.

In the discrete setting, we will often use probability model and probability distribution interchangeably with probability function. In all cases, to specify a probability function requires identifying (i) the outcomes of the sample space and (ii) the probabilities associated with those outcomes.

      Letting H denote heads and T denote tails, an obvious model for a simple coin toss is

      Actually, there is some extremely small, but nonzero, probability that a coin will land on its side. So perhaps a better model would be

      Ignoring the possibility of the coin landing on its side, a more general model is

      where . If СКАЧАТЬ