Download Discrete and Continuous Random Variables

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Chapter 7
Sec 7.1
Sample spaces need not consist of numbers. In statistics, we are most often interested in
numerical outcomes such as the "count" of an occurrence. We call X a random variable
because its values vary when the phenomenon is repeated. We use capital letters near the
end of the alphabet like X or Y.
A random variable is a variable whose value is a numerical outcome of a random
phenomenon. When a random variable describes a random phenomenon the sample
space S just lists the possible values of the random variable. There are two ways of
assigning probabilities to the values of a random variable that will dominate our
application of probability as we study statistical inference.
Random variables can be either discrete or continuous. A discrete random variable X has
a countable number of possible values. The probability distribution of X lists the values
and their probabilities in table form. The probabilities must satisfy two requirement:
1) every probability pi is a number between 0 and 1
2) p1 + p2 +,,,+pk = 1.
The probability of any event is found by adding the probabilities pi of the particular
values xi that make up the event.
In Chapters 1 and 2 we used histograms and density curves to describe finite quantitative
data. In this chapter we will use analogous methods to describe the probabilities of
discrete (finite) random variables. For discrete random variables histograms can be used
to display probability distributions instead of table form. We previously used histograms
to picture the distributions of data. The height of each bar shows the probability of the
outcome at its base. Because the heights are probabilities, they add to 1. All the bars in
the histogram have the same width so the areas of the bars also display the assignment of
probability to outcomes. See Ex. 7.1 and 7.2 (p 392-395) for more explanation.
For continuous random variables which have infinite values defined by a given interval
other methods must be employed. We cannot assign probabilities to EACH individual
value of x and then add since there are INFINITE possible values. Instead we assign
probabilities directly to events using areas under a density curve. Any density curve has
area exactly 1 underneath it, corresponding to total probability 1.
More formally...
A continuous random variable X takes all values in an interval of numbers. The
probability distribution of X is described by a density curve. The probability of any event
is the area under the density curve and above the values of X that make up the event.
The probability model for a continuous random variable assigns probabilities to
intervals of outcomes rather than to individual outcomes. In fact all continuous
probability distributions assign probability 0 to every individual outcome. Only
intervals of values have positive probability.
We ignore the distinction between > and > when finding probabilities for continuous
random variables but keep the distinction when working with discrete random variables.
Because any density curve describes an assignment of probabilities, normal distributions
are probability distributions. Recall N(mean, standard deviation) for data which
permitted standardization of data to "z scores". Random variables can also be
standardized to become a standard normal random variable (Z) having distribution N(0,1)
using the same formula. See example 7.4 (pages 400-401)