Final Exam Review II

Chapters 5-7, 9Objectives and Examples

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

Value x

ProbabilityP r[ ]x

ProductPr[ ]x x

2Pr[ ] ( )x x

= 1 = E[ ]x = Variance

and Variance

+ + +

Structure of Table:

[ ]E x n p (1 )n p p

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.

1. Write question in terms of x

2. Change x’s to z’s 3. Use the standard normal table to answer question

Steps:

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

xz

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

Chapter 5 Examples

2R and OB

1R and 1B

0R and 2B

2

1

0

(6,2) 15

(8,2) 28

C

C

(6,1) (2,1) 12

(8,2) 28

C C

C

(2,2) 1

(8,2) 28

C

C

1

30

2812

28

0

42E( )

28x

1 .5

Outcomes Valuex

ProbabilityPr[x]

Productx∙Pr[x]

[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.

[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? (1 )n p p

40 (0.75) (0.25)

7 .5

2. 74 free throws

Chapter 5 Examples

P r[ 20] ?x

20 18P r

2 .5z

μ = 18 and σ = 2.5

[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?

= 0.2119

= P r [ 0 .80] z = 1 0 .78 81

(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?

Chapter 5 Examples

The closest value is p = 0.9798 which has a corresponding Z-score of 2.05.

On the Z-table, find the Z-score that has a probability of approx. 0.9800.

18 2.05

2.5

x

xz

5.125 18x x 23 inches

Chapter 6 ObjectivesSections 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.

1 4

[1] 2 3

0 5

A

1 2 03 0 4B

Find AB and BA (if possible).

11 2 16

7 4 12

15 0 20

AB

3 2

3 32BA

Answer:

Answer:

Chapter 6 Examples

[2] 2 4 4

3 8 7 2 4 1

x y z

x y z

x y z

Solve the following system of linear equations:

Answer: 141 ; 2 ; x y z

Chapter 6 Examples

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.

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.

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

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.

. 55 .45[1 ] Le t

. 65 .35T

(b) Initially, it is 7 times as likely to be in A as B. What will the state vector be after two transitions?

2

19 13Answer :

32 32P

(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?

Answe r : 0 .585

Chapter 9 Examples

(c) Find the distribution of probabilities of A and B in the long run (i.e. find the steady state vector).

13 9Answer :

22 22P