Chapter 7 Random Variables and Probability Distributions
Objectives:
Distinguishing between discrete and continuous numeric variables
Describing variation in both discrete and continuous numeric variables by
probability distributions
Make probability statements about values of the random variable from the
distribution
Probability Distributions:
DISCRETE
CONTINUOUS
Binomial distributions
Density Curves
Geometric distributions
Normal distributions
7.1 and 7.2 Probability Distributions for Discrete Random Variables
DEFINITION
A numerical variable whose value depends on the outcome of a chance
experiment is called a random variable. A random variable associates a numerical
value with each outcome of a chance experiment.
A random variable is discrete if its set of possible values is a collection of isolated
points on the number line. The variable is continuous if its set of possible values
includes an entire interval on the number line.
Discrete
Continuous
# email messages received
#home runs per bat
# red blood cells in a sample
altitude of a plane
amount of rainfall in a city
weight of a newborn
DEFINITION
The probability distribution of a discrete random variable x gives the probability
associated with each possible x value. Each probability is the limiting relative
frequency of occurrence of the corresponding x value when the chance
experiment is repeatedly performed.Common ways to display a probability
distribution for a discrete random variable are a table, a probability histogram, or
a formula.
Review: Probability Rules
1)
2)
Discrete Random Variable X= Shirt Size
Random Variable
X = 6,7,8,9,10,11,12
Probability
X
# sold
6
85
7
122
8
138
90
154
10
177
11
133
12
92
Cumulative Probability
P(X)
1) Construct a probability distribution of the number of shirts
2) What’s the probability that a randomly selected customer will request a shirt of
size 8 or less?
3) What’s the probability that a randomly selected customer will request a shirt
that is at least an 11?
4) Construct a relative frequency histogram of the discrete probability distribution
and cumulative distribution.
x
x
The mean value of a random variable x, denoted by µ x describes where the
probability distribution of x is centered.
The standard deviation of a random variable x, denoted by σ x describes
variability in the probability distribution. When σ x is small, observed values of x
will tend to be close to the mean value (little variability). When the value of σ x is
large, there will be more variability in observed x values.
The mean value of a discrete random variable x, denoted by µ x,, is computed by
first multiplying each possible x value by the probability of observing that value
and the adding the resulting quantities. Symbolically,
µ x = ∑ x • p(x)
The term expected value is sometimes used in place of mean value, and E(x) is
alternative notation for µ x.
The variance of a discrete random variable x, denoted by σ2x is computed by first
subtracting the mean from each possible x value to obtain the deviations, then
squaring each deviation and multiplying the result by the probability of the
corresponding x value, and finally adding these quantities. Symbolically,
σ2x = ∑ (x - µ)2 • p(x)
The standard deviation of x, denoted by σ x is the square root of the variance.
5) Compute the expected shirt size of a random shopper and the standard
deviation of the shirt size.
6) I plans are to order a total of 1,000 shirts, how many shirts of size 8 should be
ordered?