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

students’ 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