Q What is a random variable?
A In many experiments the outcomes of the experiment can be assigned numerical values. For instance, if you roll a die, each outcome has a value from 1 through 6. If you ascertain the midterm test score of a student in your class, the outcome is again a number. A random variable is just a rule that assigns a number to each outcome of an experiment. These numbers are called the values of the random variable. We often use letters like X, Y and Z to denote a random variable. Here are some examples
1. Experiment: Select a mutual fund; X = the number of companies in the fund portfolio.
The values of X are 2, 3, 4, ...
2. Experiment: Select a soccer player; Y = the number of goals the player has scored during the season.
The values of Y are 0, 1, 2, 3, ...
3. Experiment: Survey a group of 10 soccer players; Z = the average number of goals scored by the players during the season.
The values of Z are 0, 0.1, 0.2, 0.3, ...., 1.0, 1.1, ...
Discrete and Continuous Random Variables
A discrete random variable can take on only specific, isolated numerical values, like the outcome of a roll of a die, or the number of dollars in a randomly chosen bank account. Discrete random variables that can take on only finitely many values (like the outcome of a roll of a die) are called finite random variables. Discrete random variables that can take on an unlimited number of values (like the number of stars estimated to be in the universe) are infinite discrete random variables.
A continuous random variable, on the other hand, can take on any values within a continuous range or an interval, like the temperature in Central Park, or the height of an athlete in centimeters.
Here are some for you:
|P(X = 4) =||
|The probability that X = 4 is 1/12|
If we tabulate the probabilities of all the possible values of X together, we get the probability distribution of X.
Click here to see the probability distribution for the above example.
You toss a coin 4 times. Three are 16 possible outcomes:
Here are some of the probabilities:
|P(X = 0) =||
|Only one of the 16 possible outcomes has X = 0; namely TTTT.|
|P(X = 1) =||
|Four of the 16 possible outcomes have X = 1; namely HTTT, THTT, TTHT, and TTTH.|
Now fill in the rest (use fractions or decimals accurate to 4 places):
Q What is the distinction between X (upper case) and x (lower case) in the table?
A The distinction is important; X stands for the random variable in question, whereas x stands for a specific value (0, 1, 2, 3, or 4) of X (so that x is always a number). Thus, if, say x = 2, then P(X = x) means P(X = 2), the probability that X is 2. Similarly, if Y is a random variable, then P(Y = y) is the probability that Y has the specific value y.
|Number of Malls||6||4||4||3||2||1|
We can take X to be the number of movie screens in a selected mall (so the values of X are given in the top row of the chart). The frequencies are then the numbers in the second row. For instance, we observed X = 0 (no movie screens) a total of 35 times, since 25 of the malls had no movie screens. So, we rewrite the table as follows:
Sometimes, values of X are not handed to us on a platter, but instead we are given ranges of values, such as income brackets. Such ranges of values are called measurement classes, as shown in this example:
The number of on-line Monday stock trades at OHaganStockTrades.com (a subsidiary of oHaganBooks.com) was measured for 50 Mondays in a row, with the following results:
(Number of Mondays)
Thus, for example, there there between 300 and 399 trades on 10 of the 50 Mondays. The measurement classes are the ranges 0-99, 100-299, etc.
We would like X to measure the number of stock trades on a randomly selected Monday,, and then calculate its probability disstribution. Since we are only given data for ranges of data, we replace each measurement class by its (rounded) midpoint. Let us take them one-by-one:
|The midpoint of the 0-99 range is its average value:||
|=||49.5, which we round to 50.|
|The midpoint of the 100-199 range is its average value:||
|=||149.5, which we round to 150.|
Now continue in the same way to obtain the remaining values of X:
Use the above frequency table to complete the (estimated) probabiliity distribution of X.
Now try some of the exercises in Section 8.1 of Finite Mathematics and Finite Mathematics and Applied Calculus