Final Exam Review II

1
Final Exam Review II Chapters 5-7, 9 Objectives
and Examples
2
Chapter 5 Objectives

• Given an experiment, compute its expected
  value, variance, or standard deviation.

If the experiment is binomial
Use Formulas
If the experiment is not binomial
Use a Probability Density Function
Structure of Table








and
3
Chapter 5 Objectives (cont.)

• Given a normal random variable x, the mean, and
  the standard deviation, find the
• probability that x is
• a) less than a certain value
• b) more than a certain value
• c) between two values.

Steps
1. Write question in terms of x 2.
Change xs to zs 3. Use the standard
normal table to answer question

• Know how to use the table when the z-score is
  off the charts.

• Know how to use the standard normal table to
  answer backwards problems.

4
Chapter 5 Examples
1 A jar contains 6 red and 2 blue marbles.
You reach in and randomly select 2
marbles. Let X represent the number of red
marbles selected. Find the expected value
of X by completing the probability table below.





Outcomes
Value x
Probability Prx
Product xPrx
2
2R and OB
1
1R and 1B
0R and 2B
0
5
Chapter 5 Examples

• 2 A basketball player makes a free throw with
  a probability of 0.75. In 40 attempts. (a)
  What is the expected number of free throws the
  basketball player will make? (b) What is the
  standard deviation?
• n 40
• (Make) p 0.75
• (Miss) 1 p 0.25
• (a) Ex np (40)(0.75)
• 30 free throws
• (b) What is the standard deviation?

free throws
6
Chapter 5 Examples
3 The annual snowfall for a city is normally
distributed with a mean of 18 inches and
standard deviation of 2.5 inches. (a) What is
the probability that the annual snowfall will
exceed 20 inches?
µ 18 and s 2.5
0.2119
(b) A city qualifies for emergency relief if
their annual snowfall is in the top 2.
How many inches of snow would need to fall
this year for the city to receive
relief?
On the Z-table, find the Z-score that has a
probability of approx. 0.9800.
The closest value is p 0.9798 which has a
corresponding Z-score of 2.05.
7
Chapter 6 Objectives
Sections 6.1/6.2

• Solving a System of Linear Equations in two or
  three variables

A system of linear equations can have one
solution, no solution, or infinitely many
solutions
Know how to use the graphing method to solve
systems of equations with two variables
Know how to use the Elimination and
Substitution methods to solve systems of
equations with two or three variables
Section 6.3

• Performing Matrix Operations

Matrix Addition/Subtraction To perform, the
matrices must have the same dimensions.
Operation is done spot-by-spot.
Scalar Multiplication Multiplying every
entry of a matrix by a constant.
Matrix Multiplication To perform, the inner
dimensions of the two matrices must be the
same (i.e. columns of the first matrix rows of
the second matrix). Operation is done by
taking linear combinations of rows and
columns.
8
Chapter 6 Examples
Find AB and BA (if possible).
Answer
Answer
9
Chapter 6 Examples
Solve the following system of linear equations
Answer
10
Chapter 7 Objectives

• Know how to solve a single linear inequality
  and determine which half plane to
• shade as the solution set (i.e. feasible
  region).

• Know how to determine whether a point is in the
  feasible region of the system of
• of inequalities.

• Know how to graph a system of linear
  inequalities, shade its feasible region, and
• identify, as ordered pairs, the corner points
  of the region.

• Know that intersection points may or may not be
  corner points of the feasible region.

• Be able to determine whether a feasible region
  is bounded or unbounded.

• Know the Fundamental Theorem of Linear
  Programming.

• Be able to find the optimal values (i.e. the
  minimum or maximum values) of a
• feasible region and the corner point(s) at
  which they occur.

Most common mistakes made on Ch 7 test questions
were?
11
Chapter 9 Objectives

• Section 9.1

• Create a transition matrix to represent a Markov
  process.

• A transition matrix always has the following
  properties

1) Same number of rows and columns
2) Every entry is a probability (0 to 1)
3) Each row (vector) sums to 1

• Use a transition matrix or a power of a
  transition matrix to answer conditional questions.

• Given the initial state vector (P0) and
  transition matrix (T), find a subsequent state
  vector (Pn). That is , find P1, P2, P3, etc. . .

Pn P0Tn

• This is done by using the formula

12
Chapter 9 Objectives

• Section 9.2

• Create a transition diagram from a transition
  matrix.

• Determine if a transition matrix is irreducible
  by looking at its transition diagram (i.e. do
  all of the states communicate in the diagram?).

• Determine if a transition matrix is regular.
• If regular, it must be irreducible and
  either contain at least one non-zero entry along
  its main diagonal, or there exists some power of
  T that makes at least one zero entry along the
  main diagonal positive.

• Find the steady state vector of a Markov chain
  (i.e. the distribution of the state probabilities
  in the long run).

• The transition matrix must be regular, or a
  steady state vector will not exist.

• The steady state vector is found by using either
  of the following equations

or
PT P
P(T I) 0
13
Chapter 9 Objectives

• Section 9.2 (cont.)

• Using the steady state vector equation,
  substitute the necessary matrices into the
  equation, perform matrix algebra (subtraction and
  multiplication), which will produce a dependent
  system of equations.

• Insert an equation (e.g. x y 1) into the
  system to produce a unique solution. Also,
  remove an equation from the system.

• Solve the remaining system of equations to find
  the steady state probabilities.

14
Chapter 9 Examples
(a) Calling the states A and B, what is the
probability that if you start in B, you will
end up in A two transitions from now?
(b) Initially, it is 7 times as likely to be in
A as B. What will the state vector be
after two transitions?
(c) Find the distribution of probabilities of A
and B in the long run (i.e. find the
steady state vector).