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AP Statistics Chapter 7 – Random Variables

7.1: Discrete and Continuous Random Variables

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

DISCRETE RANDOM VARIABLE
A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities.

The probabilities p_i must satisfy two requirements:

1. Every probability p_i is a number between 0 and 1.

2. p₁ + p₂ + … + p_k = 1

Find the probability of any event by adding the probabilities p_i of the particular values x_i that make up the event.

CONTINUOUS RANDOM VARIABLE
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.

7.2: Means and Variances of Random Variables

MEAN OF A DISCRETE RANDOM VARIABLE
Suppose that X is a discrete random variable whose distribution is

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

STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE
Suppose that X is a discrete random variable whose distribution is

and m is the mean of X. The variance of X is

and the standard deviation is the square root of the previous result.

LAW OF LARGE NUMBERS
Draw independent observations at random from any population with finite mean . As the number of observations drawn increases, the mean of the observed values eventually approaches the mean .