Review on Probability Distributions (Chapters 5 & 6)slrace/aa591/CHapter 5and6_LectureNotes.pdf ·...

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Review on Probability Distributions(Chapters 5 & 6)


Key Terms

Random Variables

Discrete vs Continuous Probability Distribution

• Properties

• Expected Value and Variance

• Cumulative Distribution Function

The Uniform & Normal Distribution


A random variable is a numerical description of theoutcome of an experiment.

Random Variables

A discrete random variable may assume either afinite number of values or an infinite sequence ofvalues.

A continuous random variable may assume anynumerical value in an interval or collection ofintervals.


Random Variables

Question Random Variable x Type

Family

size

x = Number of dependents

reported on tax returnDiscrete

Distance from

home to store

x = Distance in miles from

home to the store site

Continuous

Own dogor cat

x = 1 if own no pet;

= 2 if own dog(s) only;

= 3 if own cat(s) only;

= 4 if own dog(s) and cat(s)

Discrete


The probability distribution for a random variabledescribes how probabilities are distributed overthe values (discrete) or intervals (continuous) of a r.v.

Probability Distributions


Discrete VS Continuous Distributions

Discrete

The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable.

The required conditions for a discrete probability function are:

Continuous

It is not possible to talk about the probability for each value of the random variable. Instead, we calculate the probability for intervals.

A probability distribution or probability density function (pdf) of r.v.X is a function f (x) such that for any two numbers a and b,

f(x) > 0

f(x) = 1 𝑃 𝑎 ≤ 𝑥 ≤ 𝑏 = 𝑎

𝑏

𝑓 𝑥 𝑑𝑥 ≤ 1


a tabular representation of the probabilitydistribution for TV sales was developed.

Using past data on TV sales, …

Number

Units Sold of Days

0 80

1 50

2 40

3 10

4 20

200

x f(x)

0 .40

1 .25

2 .20

3 .05

4 .10

1.00

80/200

Discrete Probability Distributions


Continuous Probability Distributions

The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.

x

f (x)Normal

x1 x2

𝑃 𝑥1 ≤ 𝑥 ≤ 𝑥2 = 𝑃 𝑥1 < 𝑥 ≤ 𝑥2= 𝑃 𝑥1 ≤ 𝑥 < 𝑥2= 𝑃(𝑥1 < 𝑥 < 𝑥2)


The probability distribution of a random variable, if known, can be used to calculate the mean, variance, skewness, kurtosis and all other descriptive statistics of the random variable.

Using the probability distribution – f(x)


Expected Value, Variance, Std. Deviation

Discrete

Expected value, or mean:

Variance

Standard deviation

Continuous

2 = E (x - )2 = (x - )2f(x)

=E(x) = xf(x) 𝐸 𝑥 = −∞

+∞

𝑥𝑓 𝑥 𝑑𝑥

𝜎2 = −∞

+∞

𝑥 − 𝜇 2𝑓 𝑥 𝑑𝑥

Expected value, or mean:

Variance

Standard deviation

𝜎 = 𝜎2 𝜎 = 𝜎2


CUMULATIVE DISTRIBUTIONS FUNCTIONS

More on distributions…


Definition of CDF

The cumulative distribution function, F(x) for a random variable X is defined for every number x as the probability that the r.v. will take any value up to x, i.e.

𝐹 𝑥 = 𝑃(𝑋 ≤ 𝑥)

e.g. 𝐹 3 = 𝑃 𝑋 ≤ 3𝐹 −1 = 𝑃(𝑋 ≤ −1)

and so on…


CDF: Discrete vs. Continuous

The cumulative distribution function, F(x) for a discrete random variable X is defined for every number x by:

𝐹 x = 𝑃 𝑋 ≤ x =

𝑥𝑖≤𝑥

𝑓(𝑥𝑖) =

𝑥𝑖<𝑥

𝑃(𝑋 = 𝑥𝑖)

The cumulative distribution function, F(x) for a continuous random variable X is defined for every number x by:

𝐹 x = 𝑃 𝑋 ≤ x = −∞

x

𝑓 𝑥 𝑑𝑥


CDF Properties

A CDF is never decreasing (i.e. as x increases, the CDF will never decrease)

A CDF is always between 0 and 1

For a continuous random variable, the CDF, F(x), can also be used as follows:

𝑃 𝑥 ≥ 𝑎 = 1 − P x ≤ 𝑎 = 1 − 𝐹 𝑎

𝑃 𝑎 ≤ 𝑥 ≤ 𝑏 = 𝐹 𝑏 − 𝐹(𝑎)

How the above relations change if the distribution is discrete?


THE UNIFORM AND NORMAL DISTRIBUTIONS

Important continuous distributions…


Uniform Probability Distribution

where: a = smallest value the variable can assume

b = largest value the variable can assume

f (x) = 1/(b – a) for a < x < b= 0 elsewhere

A random variable is uniformly distributedwhenever the probability is proportional to the interval’s length.

The uniform probability density function is:


Var(x) = (b - a)2/12

E(x) = (a + b)/2


Expected Value of x

Variance of x



Example: Slater's Buffet

Slater customers are charged

for the amount of salad they take.

Sampling suggests that the

amount of salad taken is

uniformly distributed

between 5 ounces and 15 ounces.


Uniform Probability Density Function

f(x) = 1/10 for 5 < x < 15

= 0 elsewhere

where:

x = salad plate filling weight



Expected Value of x

Variance of x

E(x) = (a + b)/2

= (5 + 15)/2

= 10

Var(x) = (b - a)2/12

= (15 – 5)2/12

= 8.33





for Salad Plate Filling Weight

f(x)

x5 10 15

1/10

Salad Weight (oz.)


f(x)

x5 10 15

1/10

Salad Weight (oz.)

P(12 < x < 15) = 1/10(3) = .3

What is the probability that a customer

will take between 12 and 15 ounces of salad?

12



Normal Probability Distribution

The normal probability distribution is the most important distribution for describing a continuous random variable.

It is widely used in statistical inference.


Heightsof people


It has been used in a wide variety of applications:

Scientificmeasurements


Amounts

of rainfall


It has been used in a wide variety of applications:

Testscores



Normal Probability Density Function

2 2( ) /21( )

2

xf x e

= mean

= standard deviation

= 3.14159

e = 2.71828

where:


The distribution is symmetric; its skewnessmeasure is zero.


Characteristics

x


The entire family of normal probability

distributions is defined by its mean and itsstandard deviation .


Characteristics

Standard Deviation

Mean

x


The highest point on the normal curve is at themean, which is also the median and mode.


Characteristics

x



Characteristics

-10 0 20

The mean can be any numerical value: negative,zero, or positive.

x



Characteristics

= 15

= 25

The standard deviation determines the width of thecurve: larger values result in wider, flatter curves.

x


Probabilities for the normal random variable aregiven by areas under the curve. The total areaunder the curve is 1 (.5 to the left of the mean and.5 to the right).


Characteristics

.5 .5

x



Characteristics

of values of a normal random variable

are within of its mean.

68.26%

+/- 1 standard deviation



95.44%

+/- 2 standard deviations



99.72%

+/- 3 standard deviations



Characteristics

x – 3 – 1

– 2

+ 1

+ 2

+ 3

68.26%

95.44%

99.72%


Standard Normal Probability Distribution

A random variable having a normal distributionwith a mean of 0 and a standard deviation of 1 issaid to have a standard normal probabilitydistribution.


1

0

z

The letter z is used to designate the standardnormal random variable.



Converting to the Standard Normal Distribution


zx

We can think of z as a measure of the number ofstandard deviations x is from .



Example: Pep Zone

Pep Zone sells auto parts and supplies including

a popular multi-grade motor oil. When the

stock of this oil drops to 20 gallons, a

replenishment order is placed.PepZone

5w-20Motor Oil


The store manager is concerned that sales are being

lost due to stockouts while waiting for an order.

It has been determined that demand during

replenishment lead-time is normally

distributed with a mean of 15 gallons and

a standard deviation of 6 gallons.

The manager would like to know the

probability of a stockout, P(x > 20).


PepZone

5w-20Motor Oil

Example: Pep Zone


z = (x - )/

= (20 - 15)/6

= .83

Solving for the Stockout Probability

Step 1: Convert x to the standard normal distribution.

PepZone5w-20Motor Oil

Step 2: Find the area under the standard normalcurve to the left of z = .83.

see next slide



Cumulative Probability Table for

the Standard Normal Distribution

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

. . . . . . . . . . .


P(z < .83)



P(z > .83) = 1 – P(z < .83)

= 1- .7967

= .2033


Step 3: Compute the area under the standard normalcurve to the right of z = .83.


Probabilityof a stockout P(x > 20)




0 .83

Area = .7967Area = 1 - .7967

= .2033

z




If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be?


Inverse of the Standard Normal Probability Distribution


Solving for the Reorder Point


0

Area = .9500

Area = .0500

zz.05





Step 1: Find the z-value that cuts off an area of .05in the right tail of the standard normaldistribution.

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

. . . . . . . . . . .

We look up the complement of the tail area (1 - .05 = .95)





Step 2: Convert z.05 to the corresponding value of x.

x = + z.05

= 15 + 1.645(6)

= 24.87 or 25

A reorder point of 25 gallons will place the probabilityof a stockout during leadtime at (slightly less than) .05.





By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockoutdecreases from about .20 to .05.This is a significant decrease in the chance that PepZone will be out of stock and unable to meet acustomer’s desire to make a purchase.



Other Common Continuous distributions

Student’s t

• Hypothesis testing, confidence intervals, prediction intervals, modelling errors that have “heavier” tails than normal.

F

• ANOVA (Analysis of Variance), multiple hypothesis testing

Chi-square

• Hypothesis testing and ANOVA (variance estimation)

Logistic distributions

• Logistic regression

Lognormal Distribution

• Describing prices (economics) and stock market prices

Exponential Probability Distribution

• Describing the time it takes to complete a task


End of distributions review (Chapter 5 & 6)

Review on Probability Distributions (Chapters 5 & 6)slrace/aa591/CHapter 5and6_LectureNotes.pdf ·...

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