3
Discrete Random
Variables and
Probability Distributions
3.2
Probability Distributions
for Discrete Random Variables
Probability Distributions for Discrete Random
Variables
Probabilities assigned to various outcomes in in turn
determine probabilities ass
4
Continuous Random
Variables and
Probability Distributions
4.4
The Exponential Distribution
3
The Exponential Distributions
The family of exponential distributions provides probability
models that are very widely used in engineering and
science disciplin
3
Discrete Random
Variables and
Probability Distributions
3.3
Expected Values
The Expected Value of X
3
The Expected Value of X
Definition
Let X be a discrete rv with set of possible values D and pmf
p (x). The expected value or mean value of X, denoted b
4
Continuous Random
Variables and
Probability Distributions
4.1
Probability Density
Functions
Probability Density Functions
A discrete random variable (rv) is one whose possible
values either constitute a finite set or else can be listed in
an infinite se
3
Discrete Random
Variables and
Probability Distributions
3.6
The Poisson Probability
Distribution
The Poisson Probability Distribution
The binomial, hypergeometric, and negative binomial
distributions were all derived by starting with an experiment
consi
3
Discrete Random
Variables and
Probability Distributions
3.1
Random Variables
Random Variables
In general, each outcome of an experiment can be
associated with a number by specifying a rule of
association (e.g., the number among the sample of ten
compone
3
Discrete Random
Variables and
Probability Distributions
3.5
Hypergeometric and Negative
Binomial Distributions
Hypergeometric and Negative Binomial Distributions
The hypergeometric and negative binomial distributions are
both related to the binomial dis
2
Probability
2.5
Independence
Independence
The definition of conditional probability enables us to revise
the probability P(A) originally assigned to A when we are
subsequently informed that another event B has occurred;
the new probability of A is P(A |
2
Probability
2.4
Conditional Probability
Conditional Probability
The probabilities assigned to various events depend on
what is known about the experimental situation when the
assignment is made.
Subsequent to the initial assignment, partial information
2
Probability
2.3
Counting Techniques
Counting Techniques
When the various outcomes of an experiment are equally
likely (the same probability is assigned to each simple
event), the task of computing probabilities reduces to
counting.
Letting N denote the
2
Probability
2.1
Sample Spaces and Events
Sample Spaces and Events
An experiment is any activity or process whose outcome is
subject to uncertainty.
Although the word experiment generally suggests a
planned or carefully controlled laboratory testing situ
4
Continuous Random
Variables and
Probability Distributions
4.2
Cumulative Distribution
Functions and Expected Values
The Cumulative Distribution
Function
3
The Cumulative Distribution Function
The cumulative distribution function (cdf) F(x) for a discret
5
Joint Probability
Distributions and
Random Samples
5.1
Jointly Distributed
Random Variables
Two Discrete Random Variables
3
Two Discrete Random Variables
The probability mass function (pmf) of a single discrete rv X
specifies how much probability mass i
5
Joint Probability
Distributions and
Random Samples
5.2
Expected Values,
Covariance, and Correlation
Expected Values, Covariance, and Correlation
Any function h(X) of a single rv X is itself a random
variable.
However, to compute E[h(X)], it is not neces
6
Point Estimation
6.1
Some General Concepts
of Point Estimation
Some General Concepts of Point Estimation
Statistical inference is almost always directed toward
drawing some type of conclusion about one or more
parameters (population characteristics).
To
9
Inferences Based on
Two Samples
9.2
The Two-Sample t Test and
Confidence Interval
The Two-Sample t Test and Confidence Interval
Values of the population variances will usually not be
known to an investigator. In the previous section, we
illustrated for
9
Inferences Based on
Two Samples
9.5
Inferences Concerning Two
Population Variances
Inferences Concerning Two Population Variances
Methods for comparing two population variances (or
standard deviations) are occasionally needed, though such
problems arise
9
Inferences Based on
Two Samples
9.1
z Tests and Confidence Intervals
for a Difference Between
Two Population Means
z Tests and Confidence Intervals for a Difference Between Two Population Means
The inferences discussed in this section concern a
differen
8
Tests of Hypotheses
Based on a Single
Sample
8.1
Hypotheses and Test
Procedures
Hypotheses and Test Procedures
A statistical hypothesis, or just hypothesis, is a claim or
assertion either about the value of a single parameter
(population characteristic
12
Simple Linear
Regression and
Correlation
12.2
Estimating Model
Parameters
Estimating Model Parameters
We will assume in this and the next several sections that
the variables x and y are related according to the simple
linear regression model.
The value
9
Inferences Based on
Two Samples
9.3
Analysis of Paired Data
Analysis of Paired Data
We considered making an inference about a difference
between two means 1 and 2.
This was done by utilizing the results of a random sample
X1, X2,Xm from the distribution
7
Statistical Intervals
Based on a Single
Sample
7.4
Confidence Intervals for the Variance
and Standard Deviation of a Normal
Population
Confidence Intervals for the Variance and Standard Deviation of a Normal Population
Although inferences concerning a p
7
Statistical Intervals
Based on a Single
Sample
7.3
Intervals Based on a Normal
Population Distribution
Intervals Based on a Normal Population Distribution
The CI for presented in earlier section is valid provided
that n is large. The resulting interval
8
Tests of Hypotheses
Based on a
Single Sample
8.4
P-Values
P-Values
Using the rejection region method to test hypotheses
entails first selecting a significance level .
Then after computing the value of the test statistic, the null
hypothesis H0 is reject
7
Statistical Intervals
Based on a Single
Sample
7.1
Basic Properties of
Confidence Intervals
Basic Properties of Confidence Intervals
The basic concepts and properties of confidence intervals
(CIs) are most easily introduced by first focusing on a
simple