BUSN 352: Statistics Review

1
BUSN 352 Statistics Review
Professor Joseph Szmerekovsky

• Probability Distributions

2
Probability Concepts

• Let A be an event. Pr(A) is then the
  probability that A will occur
• If A never occurs, Pr(A) 0
• If A is sure to occur, Pr(A) 1

3
Example Find the probability

• Selecting a black card from the standard deck of
  52 cards
• Selecting a King
• Selecting a red King
• General Pattern Pr(A)

1/2
4/52 1/13
2/52 1/26
4
Example Find probability

• Throwing two fair dice find the probability that
• sum of the faces is equal to 2
• sum of the faces is equal to 4
• sum of the faces is equal to 7

5
Random Variables

• A random variable is a rule that assigns a
  numerical value to each possible outcome of an
  experiment.
• Discrete random variable -- countable number of
  values.
• Continuous random variable -- assumes values in
  intervals on the real line.

6
The Basics of Random Variables

• The probability distribution of a discrete
  random variable X gives the probability of each
  possible value of X.
• The sum of the probabilities must be 1.

7
Example Probability distributions

• Fair coin toss

• Two fair coins

• Find Pr(X?1)
• Find Pr(X1.5)
• Find Pr(X?1.5)

8
Expected Value

• The expected value (mean) of the random variable
  is the sum of the products of x and the
  corresponding probabilities

9
Example Insurance Policy

• Alice sells Ben a 10,000 insurance policy at an
  annual premium of 460.
• If Pr(Ben dies next year) .002, what is the
  expected profit of the policy?

E(X) 460(.998) (-9540)(.002) 440
10
Example Debbon Air Seat Release

• Debbon Air needs to make a decision about Flight
  206 to Myrtle Beach.
• 3 seats reserved for last-minute customers (who
  pay 475 per seat), but the airline does not know
  if anyone will buy the seats.
• If they release them now, they know they will be
  able to sell them all for 250 each.

11
Debbon Air Seat Release

• The decision must be made now, and any number of
  the three seats may be released.
• Debbon Air counts a 150 loss of goodwill for
  every last-minute customer turned away.
• Probability distribution for X of
  last-minute customers requesting seats

12
Debbon Air Seat Release

• What is Debbon Airs expected net revenue
  (revenue minus loss of goodwill) if all three
  seats are released now?

• X 0 Net Revenue 3(250) 750

• X 1 Net Rev 3(250) - 150 600

E (Net Revenue)
750(.45) 600(.30) 450(.15) 300(.10)
615.
13
Debbon Air Seat Release

• How many seats should be released to maximize
  expected net revenue?

Two seats should be released.
14
Variance and Standard Deviation of Random
Variables

• The variance of a discrete R.V. X is

• The standard deviation is the square root of the
  variance.

15
Continuous random variables

• Continuous random variable -- assumes values in
  intervals on the real line.

Total area 1
16
Example Uniform distribution

• Is this a valid probability density function?

Yes

• Find Pr(0.2 lt X lt 0.5)

0.31 0.3

• Find Pr(X gt 0.6)

0.41 0.4
17
The Normal Probability Model

• Importance of the Normal model
• Numerous phenomena seem to follow it, or can be
  approximated by it.
• It provides the basis for classical statistical
  inference through the Central Limit Theorem.
• It motivates the Empirical Rule.

18
The Normal Probability Model

• Crucial Properties
• Bell-shaped, symmetric
• Measures of central tendency (mean, median) are
  the same.
• Parameters are mean and standard deviation
  .

19
The Normal Probability Model

• The Normal probability density function

The Bell Curve
fY(y)
y
20
The Normal Probability Model
This area
0.5
This area
fY(y)
y
a
b
21
The Normal Probability Model Effect of the Mean
22
The Normal Probability Model Effect of the SD
2
1
23
The Normal Probability Model and Empirical Rule
24
The Standard Normal Distribution
Normal with Mean SD
Standard Normal with Mean 0 and SD 1
0
1
2
-1
-2
25
Table A.1 Standard Normal Distribution

• Standard Normal random variable Z
• E(Z) 0 and SD(Z) 1
• Table A.1 gives Standard Normal probabilities to
  four decimal places.

.4332
fZ(z)
z
0
1.50
26
(No Transcript)
27
Practice with Table A.1
.5 - .4332
.0668
Pr(Z gt 0) .5
fZ(z)
.4332
z
0
1.50
28
Practice with Table A.1
.4332 .5
.9332
Pr(Z lt 0) .5
.4332
1.5
0
29
Practice with Table A.1
So k is about 1.645
.4500
.4495
k
0
1.64
30
Z Scores Standardizing Normal Distributions

• Suppose X is
• Transformation Formula

• For a given x, the Z score is the number of SDs
  that x lies away from the mean.

31
Example Tele-Evangelist Donations

• Money collected daily by a tele-evangelist,
  Y,is Normal with mean 2000, and SD 500.

• What is the chance that tomorrows donations
  will be less than 1500?

Convert to Z scores
32
Tele-Evangelist Donations

• Money collected is Normal with mean 2000 and SD
  500.
• What is the probability that tomorrows
  donations are between 2000 and 3000?

• Let Y collected tomorrow
• Y is Normal with mean 2000 and SD 500
• Need

• Convert to Z scores

.4772
33
Tele-Evangelist Donations

• What is the chance that tomorrows donations
  will exceed 3000?

• Y is still Normal with mean 2000 and SD 500...

Convert to Z scores