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Professor Joseph Szmerekovsky. BUSN 352: Statistics Review. Probability Distributions. 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. Example: Find the probability.

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#### Presentation Transcript

Professor Joseph Szmerekovsky

## BUSN 352: Statistics Review

Probability Distributions

### 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

### 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

### 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

### 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.

### 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.

### Example: Probability distributions

• Fair coin toss

• Two fair coins

• Find Pr(X1)

• Find Pr(X=1.5)

• Find Pr(X1.5)

### Expected Value

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

### 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

### 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.

### 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:

### “Debbon Air” Seat Release

• What is Debbon Air’s 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.

### “Debbon Air” Seat Release

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

Two seats should be released.

### 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.

Probability density function

f(x)

Area under the graph = Pr(a<X<b)

a

x

b

### Continuous random variables

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

Total area = 1

f(x)

1

x

0.2

0.5

0.6

1

### Example: Uniform distribution

• Is this a valid probability density function?

Yes

• Find Pr(0.2 < X < 0.5)

0.3·1 = 0.3

• Find Pr(X > 0.6)

0.4·1 = 0.4

### 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.

### The Normal Probability Model

• Crucial Properties

• Bell-shaped, symmetric

• Measures of central tendency (mean, median) are the same.

• Parameters are mean and standard deviation .

### The Normal Probability Model

The Normal probability density function:

“The Bell Curve”

fY(y)

y

This area =

0.5

This area =

fY(y)

y

a

b

2

1

Normal with

Mean

SD

Standard Normal

with Mean 0

and SD 1

-2

-1

0

+1

+2

### 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

= .5 - .4332

= .0668

Pr(Z > 0) = .5

fZ(z)

.4332

z

0

1.50

= .4332 + .5

= .9332

Pr(Z < 0) = .5

.4332

0

1.5

.4500

.4495

k

0

1.64

### Z Scores: Standardizing Normal Distributions

• Suppose X is

• Transformation Formula:

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

### 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 tomorrow’s donations will be less than \$1500?

Convert to Z scores

### Tele-Evangelist Donations

• Money collected is Normal with mean \$2000 and SD \$500.

• What is the probability that tomorrow’s 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

### Tele-Evangelist Donations

• What is the chance that tomorrow’s donations will exceed \$3000?

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

Convert to Z scores