1 1 Slide 2004 Thomson/South-Western Chapter 3, Part A Discrete
Probability Distributions n Random Variables n Discrete Probability
Distributions n Expected Value and Variance n Binomial Probability
Distribution n Poisson Probability Distribution.10.20.30.40 0 1 2 3
4 Slide 2 2 2 Slide 2004 Thomson/South-Western A random variable is
a numerical description of the A random variable is a numerical
description of the outcome of an experiment. outcome of an
experiment. A random variable is a numerical description of the A
random variable is a numerical description of the outcome of an
experiment. outcome of an experiment. Random Variables A discrete
random variable may assume either a A discrete random variable may
assume either a finite number of values or an infinite sequence of
finite number of values or an infinite sequence of values. values.
A discrete random variable may assume either a A discrete random
variable may assume either a finite number of values or an infinite
sequence of finite number of values or an infinite sequence of
values. values. A continuous random variable may assume any A
continuous random variable may assume any numerical value in an
interval or collection of numerical value in an interval or
collection of intervals. intervals. A continuous random variable
may assume any A continuous random variable may assume any
numerical value in an interval or collection of numerical value in
an interval or collection of intervals. intervals. Slide 3 3 3
Slide 2004 Thomson/South-Western Expected Value and Variance The
expected value, or mean, of a random variable The expected value,
or mean, of a random variable is a measure of its central location.
is a measure of its central location. The expected value, or mean,
of a random variable The expected value, or mean, of a random
variable is a measure of its central location. is a measure of its
central location. The variance summarizes the variability in the
The variance summarizes the variability in the values of a random
variable. values of a random variable. The variance summarizes the
variability in the The variance summarizes the variability in the
values of a random variable. values of a random variable. The
standard deviation, , is defined as the positive The standard
deviation, , is defined as the positive square root of the
variance. square root of the variance. The standard deviation, , is
defined as the positive The standard deviation, , is defined as the
positive square root of the variance. square root of the variance.
Var( x ) = 2 = ( x - ) 2 f ( x ) E ( x ) = = xf ( x ) Slide 4 4 4
Slide 2004 Thomson/South-Western n Four Properties of a Binomial
Experiment 3. The probability of a success, denoted by p, does not
change from trial to trial. not change from trial to trial. 3. The
probability of a success, denoted by p, does not change from trial
to trial. not change from trial to trial. 4. The trials are
independent. 2. Two outcomes, success and failure, are possible on
each trial. on each trial. 2. Two outcomes, success and failure,
are possible on each trial. on each trial. 1. The experiment
consists of a sequence of n identical trials. identical trials. 1.
The experiment consists of a sequence of n identical trials.
identical trials. stationarityassumption Binomial Probability
Distribution Slide 5 5 5 Slide 2004 Thomson/South-Western Poisson
Probability Distribution n Two Properties of a Poisson Experiment
2. The occurrence or nonoccurrence in any interval is independent
of the occurrence or interval is independent of the occurrence or
nonoccurrence in any other interval. nonoccurrence in any other
interval. 2. The occurrence or nonoccurrence in any interval is
independent of the occurrence or interval is independent of the
occurrence or nonoccurrence in any other interval. nonoccurrence in
any other interval. 1. The probability of an occurrence is the same
for any two intervals of equal length. for any two intervals of
equal length. 1. The probability of an occurrence is the same for
any two intervals of equal length. for any two intervals of equal
length. Slide 6 6 6 Slide 2004 Thomson/South-Western Chapter 3,
Part B Continuous Probability Distributions n Uniform Probability
Distribution n Normal Probability Distribution n Exponential
Probability Distribution f ( x ) x x Uniform x Normal x x
Exponential Slide 7 7 7 Slide 2004 Thomson/South-Western Normal
Probability Distribution n The normal probability distribution is
the most important distribution for describing a continuous random
variable. n It is widely used in statistical inference. Slide 8 8 8
Slide 2004 Thomson/South-Western Heights of people Heights Normal
Probability Distribution n It has been used in a wide variety of
applications: Scientific measurements measurementsScientific Slide
9 9 9 Slide 2004 Thomson/South-Western The distribution is
symmetric, and is bell-shaped. The distribution is symmetric, and
is bell-shaped. Normal Probability Distribution n Characteristics x
Slide 10 10 Slide 2004 Thomson/South-Western The entire family of
normal probability The entire family of normal probability
distributions is defined by its mean and its distributions is
defined by its mean and its standard deviation . standard deviation
. The entire family of normal probability The entire family of
normal probability distributions is defined by its mean and its
distributions is defined by its mean and its standard deviation .
standard deviation . Normal Probability Distribution n
Characteristics Standard Deviation Mean x Slide 11 11 Slide 2004
Thomson/South-Western The highest point on the normal curve is at
the The highest point on the normal curve is at the mean, which is
also the median and mode. mean, which is also the median and mode.
The highest point on the normal curve is at the The highest point
on the normal curve is at the mean, which is also the median and
mode. mean, which is also the median and mode. Normal Probability
Distribution n Characteristics x Slide 12 12 Slide 2004
Thomson/South-Western Normal Probability Distribution n
Characteristics = 15 = 25 The standard deviation determines the
width of the curve: larger values result in wider, flatter curves.
The standard deviation determines the width of the curve: larger
values result in wider, flatter curves. x Slide 13 13 Slide 2004
Thomson/South-Western Probabilities for the normal random variable
are Probabilities for the normal random variable are given by areas
under the curve. The total area given by areas under the curve. The
total area under the curve is 1 (.5 to the left of the mean and
under the curve is 1 (.5 to the left of the mean and.5 to the
right)..5 to the right). Probabilities for the normal random
variable are Probabilities for the normal random variable are given
by areas under the curve. The total area given by areas under the
curve. The total area under the curve is 1 (.5 to the left of the
mean and under the curve is 1 (.5 to the left of the mean and.5 to
the right)..5 to the right). Normal Probability Distribution n
Characteristics.5.5 x Slide 14 14 Slide 2004 Thomson/South-Western
Normal Probability Distribution n Characteristics of values of a
normal random variable of values of a normal random variable are
within of its mean. are within of its mean. of values of a normal
random variable of values of a normal random variable are within of
its mean. are within of its mean.68.26%68.26% +/- 1 standard
deviation of values of a normal random variable of values of a
normal random variable are within of its mean. are within of its
mean. of values of a normal random variable of values of a normal
random variable are within of its mean. are within of its mean.
95.44%95.44% +/- 2 standard deviations of values of a normal random
variable of values of a normal random variable are within of its
mean. are within of its mean. of values of a normal random variable
of values of a normal random variable are within of its mean. are
within of its mean.99.72%99.72% +/- 3 standard deviations Slide 15
15 Slide 2004 Thomson/South-Western Normal Probability Distribution
n Characteristics x 3 1 2 + 1 + 2 + 3 68.26% 95.44% 99.72% Slide 16
16 Slide 2004 Thomson/South-Western 0 z The letter z is used to
designate the standard The letter z is used to designate the
standard normal random variable. normal random variable. The letter
z is used to designate the standard The letter z is used to
designate the standard normal random variable. normal random
variable. Standard Normal Probability Distribution Slide 17 17
Slide 2004 Thomson/South-Western n Converting to the Standard
Normal Distribution Standard Normal Probability Distribution We can
think of z as a measure of the number of standard deviations x is
from . Slide 18 18 Slide 2004 Thomson/South-Western n Probability
Table for the Standard Normal Distribution Example: Pep Zone Pep
Zone 5w-20 Motor Oil P (0 < z
