ProbabilityStudy Guide for Section 6.8

Note: We are only covering basic probability distributions and the normal distribution. You do notneed to worry about average values or the exponential distribution.

1. Random Variables and Probability Distributions

A is any measurement whose value includes an element of randomness. Anyrandom variablerandom variable has a (or ) that\ 0 Bprobability distribution probability density function a brepresents the relative likelihood of different values for . For example, the following\distribution shows a random variable that takes on values between and , with values between" &" $ $ & and being much more likely than values between and :

B

C

1 2 3 4 5 6

You can find the probability that lies in a certain range by finding the area in that range under\the graph of :0 Ba b

T + \ , 0 B .Ba b a b(+

,

Note that .the total area under a probability distribution must be "

Problems: 3, 8

2. Normal Distributions

The most common kind of probability distribution is the (or ):normal distribution bell curve

.

5 5

The central (or average) value of the distribution is called the , and is denoted by the Greekmeanletter (mu). The width of the distribution is called the , and is denoted. standard deviationby the Greek letter (sigma). The formula for a normal distribution with mean and standard5 .deviation is:5

0 B /"

#a b 5 1

B "## #a b. 5

The simlest case is when and . This is called the :. 5 ! " standard normal distribution

0 B /"

#a b 1

B"##

Problems: 13, 15, 17