Download Chapters 5

Final Exam
Review II
Chapters 5-7, 9
Objectives and Examples
Chapter 5 Objectives
 Given an experiment, compute its expected value, variance, or standard deviation.
If the experiment is binomial
Use Formulas:
E[ x]  n  p
If the experiment is not binomial
  n  p  (1  p)
Use a Probability Density Function
Structure of Table:
Probability
P r [ x]
Value
x
Product
x  P r [ x]
+
+
= 1
Pr[ x]  ( x   )2
+
= E[ x]
= Variance
and   Variance
Chapter 5 Objectives (cont.)
 Given a normal random variable x, the mean, and the standard deviation, find the
probability that x is:
a)
b)
c)
less than a certain value
more than a certain value
between two values.
Steps:
1. Write question in terms of x
2. Change x’s to z’s
3. Use the standard normal table to answer question
z
x

 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.
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
2R and OB
2
1R and 1B
1
0R and 2B
0
Probability
Pr[x]
Product
x∙Pr[x]
C (6, 2) 15

C (8, 2) 28
C (6,1)  C (2,1) 12

C (8, 2)
28
C (2, 2) 1

C (8, 2) 28
30
28
12
28
1
0
E( x ) 
 1.5
42
28
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) E[x] = n∙p = (40)∙(0.75)
= 30 free throws
(b) What is the standard deviation?   n  p  (1  p)

40  (0.75)  (0.25)

7. 5
 2 . 7 4 free throws
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 σ = 2.5
P r [ x  2 0]  ?
2 0 1 8 

Pr z 
2 . 5 

= P r [ z  0 . 8 0]
= 1  0.78 81
=
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.
x
z

x  18
 2.05 
2.5
 5.125  x  18  x  23
inches
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.
Chapter 6 Examples
[1]
 1 4 
A   2  3
 0 5 
B  1 2 0 
3 0 4 
Find AB and BA (if possible).
Answer:
1 1  2 1 6 
A B    7 4  1 2 
 1 5 0
2 0 
Answer:
 3  2
BA  

 3 3 2 
Chapter 6 Examples
Solve the following system of linear equations:
[2]
x  2 y  4z   4
 3x  y  8 z  7
2 x  y  4z   1
Answer:
x  1
;
y2
;
z
1
4
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 it’s 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…?
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. . .
 This is done by using the formula:
Pn = P0∙Tn
Chapter 9 Objectives
 Section 9.2
 Create a transition diagram from a transition matrix.
 Determine if a transition matrix is irreducible by looking at it’s 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 it’s 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:
PT = P
or
P(T – I) = 0
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.
Chapter 9 Examples
. 5 5 . 4 5 
[ 1 ] Le t T  

.
6
5
.
3
5


(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?
Answer:
0.585
(b) Initially, it is 7 times as likely to be in A as B. What will the state vector be
after two transitions?
1 9 1 3 
An s w e r :
P2  
 3 2 3 2 
(c) Find the distribution of probabilities of A and B in the long run (i.e. find the
steady state vector).
13 9 
An s w e r :
P
 2 2 2 2 