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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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probability concepts
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
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
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
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 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
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
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
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
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
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 release1
“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 release2
“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
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.
continuous random variables

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

example uniform distribution

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
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 model1
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 model2
The Normal Probability Model

The Normal probability density function:

“The Bell Curve”

fY(y)

y

the normal probability model3
The Normal Probability Model

This area =

0.5

This area =

fY(y)

y

a

b

the standard normal distribution
The Standard Normal Distribution

Normal with

Mean

SD

Standard Normal

with Mean 0

and SD 1

-2

-1

0

+1

+2

table a 1 standard normal distribution
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

practice with table a 1
Practice with Table A.1

= .5 - .4332

= .0668

Pr(Z > 0) = .5

fZ(z)

.4332

z

0

1.50

practice with table a 11
Practice with Table A.1

= .4332 + .5

= .9332

Pr(Z < 0) = .5

.4332

0

1.5

practice with table a 12
Practice with Table A.1

So k is about 1.645

.4500

.4495

k

0

1.64

z scores standardizing normal distributions
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
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
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 donations1
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