Business and Finance College Principles of Statistics Eng. Heba Hamad 2008

Business and Finance College

Principles of Statistics

Eng. Heba Hamad2008

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSSt. Edward’s UniversitySt. Edward’s University

Continuous Probability Distributions

Chapter 6

Continuous Probability Distributions• Uniform Probability Distribution• Normal Probability Distribution• Exponential Probability Distribution

f (x)f (x)

x x

UniformUniform

xx

f f ((xx)) NormalNormalxx

f (x)f (x) ExponentialExponential

Continuous Probability Distributions

• A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.

• It is not possible to talk about the probability of the random variable assuming a particular value.

• Instead, we talk about the probability of the random variable assuming a value within a given interval.

Continuous Probability DistributionsContinuous 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.

f (x)f (x)

x x

UniformUniform

xx11 xx11 xx22 xx22

xx

f f ((xx)) NormalNormal

xx11 xx11 xx22 xx22

xx11 xx11 xx22 xx22

ExponentialExponential

xx

f (x)f (x)

xx11

xx11

xx22 xx22

Uniform Probability Distribution

where: where: aa = smallest value the variable can assume = smallest value the variable can assume

bb = largest value the variable can assume = largest value the variable can assume

f f ((xx) = 1/() = 1/(bb – – aa) for ) for aa << xx << bb = 0 elsewhere= 0 elsewhere

• A random variable is uniformly distributed whenever the probability is proportional to the interval’s length.

• The uniform probability density function is:

Var(Var(xx) = () = (bb - - aa))22/12/12

E(E(xx) = () = (aa + + bb)/2)/2

Uniform Probability Distribution

• Expected Value of x

• Variance of x

Uniform Probability Distribution

• Example: Slater's Buffet

Slater customers are chargedfor the amount of salad they take. Sampling suggests that theamount of salad taken is uniformly distributedbetween 5 ounces and 15 ounces.

Uniform Probability Density FunctionUniform Probability Density Function

ff((xx) = 1/10 for 5 ) = 1/10 for 5 << xx << 15 15

= 0 elsewhere= 0 elsewhere

where:where:

xx = salad plate filling weight = salad plate filling weight

Uniform Probability Distribution

Expected Value of Expected Value of xx

Variance of Variance of xx

E(E(xx) = () = (aa + + bb)/2)/2

= (5 + 15)/2= (5 + 15)/2

= 10= 10

Var(Var(xx) = () = (bb - - aa))22/12/12

= (15 – 5)= (15 – 5)22/12/12

= 8.33= 8.33

Uniform Probability Distribution

• Uniform Probability Distributionfor Salad Plate Filling Weight

f(x)f(x)

x x55 1010 1515

1/101/10

Salad Weight (oz.)Salad Weight (oz.)

Uniform Probability Distribution

f(x)f(x)

x x55 1010 1515

1/101/10

Salad Weight (oz.)Salad Weight (oz.)

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

What is the probability that a customerWhat is the probability that a customer

will take between 12 and 15 ounces of will take between 12 and 15 ounces of salad?salad?

1212

Uniform Probability Distribution

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.

HeightsHeightsof peopleof people

Normal Probability DistributionNormal Probability Distribution

It has been used in a wide variety of It has been used in a wide variety of applications:applications:

ScientificScientific measurementsmeasurements

AmountsAmounts

of rainfallof rainfall

Normal Probability DistributionNormal Probability Distribution

It has been used in a wide variety of It has been used in a wide variety of applications:applications:

TestTest scoresscores

Normal Probability Distribution

• Normal Probability Density Function

2 2( ) / 21( )

2xf x e

= mean= mean

= standard deviation= standard deviation

= 3.14159= 3.14159

ee = 2.71828 = 2.71828

where:where:

The distribution is The distribution is symmetricsymmetric; its skewness; its skewness measure is zero.measure is zero.

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

xx

The entire family of normal probabilityThe entire family of normal probability distributions is defined by itsdistributions is defined by its meanmean and its and its standard deviationstandard deviation . .

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

Standard Deviation Standard Deviation

Mean Mean xx

The The highest pointhighest point on the normal curve is at the on the normal curve is at the meanmean, which is also the , which is also the medianmedian and and modemode..

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

xx

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

-10-10 00 2020

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

xx

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

= 15= 15

= 25= 25

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

xx

Probabilities for the normal random variable areProbabilities for the normal random variable are given by given by areas under the curveareas under the curve. The total area. The total area under the curve is 1 (.5 to the left of the mean andunder the curve is 1 (.5 to the left of the mean and .5 to the right)..5 to the right).

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

.5.5 .5.5

xx

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

of values of a normal random variableof values of a normal random variable are within of its mean.are within of its mean.68.26%68.26%

+/- 1 standard deviation+/- 1 standard deviation

of values of a normal random variableof values of a normal random variable are within of its mean.are within of its mean.95.44%95.44%

+/- 2 standard deviations+/- 2 standard deviations

of values of a normal random variableof values of a normal random variable are within of its mean.are within of its mean.99.72%99.72%

+/- 3 standard deviations+/- 3 standard deviations

Normal Probability DistributionNormal Probability Distribution

CharacteristicsCharacteristics

xx – – 33 – – 11

– – 22 + 1+ 1

+ 2+ 2 + 3+ 3

68.26%68.26%95.44%95.44%99.72%99.72%