Discrete and Continuous Random Variables:

 

A variable is a quantity whose value changes. 

 

A discrete variable is a variable whose value is obtained by counting.

 

Examples:     number of students present

                                    number of red marbles in a jar

                                    number of heads when flipping three coins

                                    studentsí grade level

 

A continuous variable is a variable whose value is obtained by measuring.

 

Examples:     height of students in class

                        weight of students in class

                        time it takes to get to school

                        distance traveled between classes

 

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

 

         A random variable is denoted with a capital letter

         The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values

         A random variable can be discrete or continuous

 

A discrete random variable X has a countable number of possible values.

 

Example: Let X represent the sum of two dice.

 

 

Then the probability distribution of X is as follows:

 

X

2

3

4

5

6

7

8

9

10

11

12

P(X)

 

 

To graph the probability distribution of a discrete random variable, construct a probability histogram.

 

 

 

A continuous random variable X takes all values in a given interval of numbers.

 

         The probability distribution of a continuous random variable is shown by a density curve.

         The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints

         The probability that a continuous random variable X is exactly equal to a number is zero

 


 

Means and Variances of Random Variables:

 

The mean of a discrete random variable, X, is its weighted average.  Each value of X is weighted by its probability.

 

To find the mean of X, multiply each value of X by its probability, then add all the products.

 

 

The mean of a random variable X is called the expected value of X.

 

 

Law of Large Numbers:

 

As the number of observations increases, the mean of the observed values,  , approaches the mean of the population, .

 

The more variation in the outcomes, the more trials are needed to ensure that  is close to .

 

 

 

Rules for Means:

 

If X is a random variable and a and b are fixed numbers, then

 

 

If X and Y are random variables, then

 

 

 

Example:

Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y.  Suppose the average PSAT math score is 48.  What is the average SAT math score?

 

                       

 

 

 

 

Example:

Let  represent the average SAT math score.

Let  represent the average SAT verbal score.

 

 represents the average combined SAT score.  Then  is the average combined total SAT score.

 

 

 

The  Variance of a Discrete Random Variable:

 

If X is a discrete random variable with mean , then the variance of X is

 

 

The standard deviation  is the square root of the variance.

 

 

 

 

Rules for Variances:

 

If X is a random variable and a and b are fixed numbers, then

 

 

If X and Y are independent random variables, then

                       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example:

Suppose the equation Y = 20 + 100X converts a PSAT math score, X, into an SAT math score, Y.  Suppose the standard deviation for the PSAT math score is 1.5 points.  What is the standard deviation for the SAT math score?

 

 

 

Suppose the standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points.  What is the standard deviation for the combined SAT score?

 

*** Because the SAT math score and SAT verbal score are not independent, the rule for adding variances does not apply!