AP Statistics – Chapter 7 – Random Variables – Notes
7.1 Discrete & Continuous Random Variables
I. Discrete Random Variables
a. A random variable is a variable whose value is numerical outcome of a random
phenomenon (usually denoted by capital letters, X and Y.)
b. A discrete random variable X has a countable number of possible values. The probability
distribution of a discrete random variable X lists the values and their probabilities.
c. The probabilities must be…
i. Every probability, p, must be a number between 0 and 1.
ii. The sum of the probabilities is 1.
d. You can find the probability of any event by adding the probabilities, p, of the particular
values x that make up the event.
e. X = the number of ________. (discrete/countable)
II. Continuous Random Variable
a. A continuous random variable has a non-countable number of possible events (its values
are not isolated numbers but an entire interval of numbers.).
b. 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.
c. All continuous probability distribution assign probability of 0 to every individual outcome.
d. Normal Distributions are continuous random variable probability distributions.
e. X = the amount of ________. (continuous/non-countable)
7.2 Means & Variances of Random Variables
III. Mean & Variance of a Random Variable
a. Mean of a probability distribution = expected value =  X   xi P( xi ) .
b. To find the mean of X, multiply the values by its probability, then add all the products.
c. The variance of X is  2X   ( xi   X ) 2 P( xi )
d. The standard deviation  X 
(x  
i
X
) 2 P( xi ) .
IV. Law of Large Numbers
a. Draw independent observations at random from any population with finite mean  X . Decide
how accurately you would like to estimate  X . As the number of observations drawn
increases, the mean ( x ) of the observed values eventually approaches the mean u of the
population as closely as you specified and then stays that close.
V.
Rules for Means
a. Rule 1: If X is a random variable and a and b are fixed numbers, then  a bX  a  b X .
b. Rule 2: If X and Y are random variables, then  X Y   X   Y .
VI.
Rules for Variances
a. Rule 1: If X is a random variable and a and b are fixed numbers, then  2a bX  b 2 2X .
b. Rule 2: : If X and Y are independent random variables, then  2X Y   2X   Y2 .
VII. Normal Random Variables
a. Any linear combination of independent Normal random variables is also Normally
distributed.