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Random Variables

Recall that a probability assigns numbers to events. But in many problems there are only a few events of interest and, furthermore, they can often be characterized in terms of a variable.

For example, in the first roll in the game of craps (roll a pair of dice) the events of interest are: the sum of upfaces is 2, or 3, or 4, ... , or 12. Hence, there are only 11 events of interest. If we let

X = the sum of the upfaces
then the events of interest can be expressed as: X=2, X=3, ..., or X=12. Hence, X characterizes the events of interest. We call X a random variable .

As another problem, reconsider the urn problem  where the urn contained 30 blue balls and 50 red balls and that 3 of these balls were selected at random without replacement. Recall we wanted to determine the probability that all the balls were of the same color. Just let

X = the number of blue balls in the sample of size 3
then the event of interest is X =0 or X = 3. Hence, X characterizes the events of interest.

The range of a random variable is the set of values it can assume. For example in the game of craps, the range of X is {2, 3, ... ,12} while in the urn problem, the range of X is {0, 1, 2, 3}. As another example, let X be the height of an adult male in inches. It is hard, even, impossible to come up with minimum or maximum of X ; hence, a convenient range is the interval $(0, \infty)$. This seems odd at first, but keep in mind we are trying to model height. Actually the best model of height employs a range of $(- \infty, \infty)$. We will discuss this later.

Essentially, random variables come in two types: discrete and continuous random variables . A discrete random variable  has a finite (or listable) range. The range of a continuous random variable is an interval of numbers. In the first two examples, the random variables are discrete while in the last example on height, the random variable is continuous.


Exercise 5.1.1  
1.
Let X denote the number of aces in a 2 card hand drawn without replacement fro a standard deck 0f 52 cards. What is the range of X? Is it discrete or continuous?
2.
In the last problem, let Y denote the average area of the two dealt cards. Assume that we can measure area infinitely precise. What is the range of Y? Is it discrete or continuous?
3.
In the urn problem discussed above, let Z denote the number of red balls in the sample (without replacement) of size 3. What is the range of Z? Is it discrete or continuous?
4.
Let X denote the temperature at noon in Kalamazoo in centigrade. What is the range of X? Is it discrete or continuous?
5.
Let X denote the number of people in a queue at a bank teller's window. What is the range of X? Is it discrete or continuous?


next up previous contents index
Next: Discrete Populations (Probability Models) Up: Discrete Populations (Probability Models) Previous: Discrete Populations (Probability Models)

2001-01-01