Continuous Probability Distributions

• Uniform Probability Distribution• Area as a measure of Probability• The Normal Curve• The Standard Normal Distribution• Computing Probabilities for a Standard Normal

Distribution

f(x)

X

Uniform Probability Distribution

Chicago

NY

•Consider the random variable x representing the flight time of an airplane traveling from Chicago to NY.

•Under normal conditions, flight time is between 120 and 140 minutes.

•Because flight time can be any value between 120 and 140 minutes, x is a continuous variable.

Uniform Probability Distribution

140120for x

elsewhere 0

020

1)(xf

With every one-minute interval being equally

likely, the random variable x is said to have

a uniform probability distribution

elsewhere

0

1)( abxf

Uniform Probability Distribution

bxa for

elsewhere

For the flight-time random variable,

a = 120 and b = 140

Uniform Probability Density Function for Flight time

20

1

)(xf

120 140125 130 135 x

The shaded area indicates the probability

the flight will arrive in the interval between 120 and 140 minutes

Basic Geometry

Remember when we multiply a line

segment times a line segment, we get an

area

Probability as an Area

20

1

)(xf

x120 140125 130 135

Question: What is the probability that arrival time will be between 120 and 130 minutes—that is:

)130120( xP

10

50.20/10)10(20/1Area)130120( xP

Notice that in the continuous case we do not talk of a

random variable assuming a specific value. Rather, we talk of the probability that a random

variable will assume a value within a given interval.

0)( area,by measured is When x

E(x) and Var(x) for the Uniform Continuous Distribution

2)(

baxE

12

)()(

2abXVar

1302

)140120()(

xE

Applying these formulas to the example of flight times of Chicago to NY, we have:

3.3312

)120140()(

2

XVar

minutes 77.533.33 Thus

Normal Probability Distribution

The normal distribution is by far the most important

distribution for continuous random variables. It is widely

used for making statistical inferences in both the natural

and social sciences.

HeightsHeightsof peopleof peopleHeightsHeights

of 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

ScientificScientific measurementsmeasurements

AmountsAmounts

of rainfallof rainfall

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 scoresscoresTestTest

scoresscores

The Normal Distribution

22 2/)(

2

1)(

xexf

Where:

μ is the mean

σ is the standard deviation

= 3.1459

e = 2.71828

The distribution is The distribution is symmetricsymmetric, and is , and is bell-shapedbell-shaped.. The distribution is The distribution is symmetricsymmetric, and is , and is bell-shapedbell-shaped..

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 . .

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

MeanMean 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.. 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. 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.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).

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

The Standard Normal Distribution

0

The Standard Normal Distribution is a normal distribution with the special properties that is mean is zero and

its standard deviation is one.

1

00zz

The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable. The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable.

Standard Normal Probability DistributionStandard Normal Probability Distribution

Cumulative ProbabilityCumulative Probability

00 11zz

)1( zP

Probability that z ≤ 1 is the area under the curve to the left of 1.

What is P(z ≤ 1)?

Z .00 .01 .02

●

●

●

.9 .8159 .8186 .8212

1.0 .8413 .8438 .8461

1.1 .8643 .8665 .8686

1.2 .8849 .8869 .8888

●

●

To find out, use the Cumulative Probabilities Table for the Standard Normal Distribution

)1( zP

Exercise 1

2.46

a) What is P(z ≤2.46)?

b) What is P(z ≥2.46)?

Answer:

a) .9931

b) 1-.9931=.0069

z

Exercise 2

-1.29

a) What is P(z ≤-1.29)?

b) What is P(z ≥-1.29)?

Answer:

a) 1-.9015=.0985

b) .9015

Note that, because of the symmetry, the area to the left of -1.29 is the same as the area to the right of 1.29

1.29

Red-shaded area is equal to green- shaded area

Note that:

)29.1(1)29.1( zPzP

z

Exercise 3

0

What is P(.00 ≤ z ≤1.00)?

1

3413.5000.8413.

)0()1()100(.

zPzPzP

P(.00 ≤ z ≤1.00)=.3413

z

Exercise 4

0

What is P(-1.67 ≥ z ≥ 1.00)?

1

7938.)9525.1(8413.

)]67.1(1[)1()167.1(

zPzPzP

P(-1.67 ≤ z ≤1.00)=.7938

-1.67

Thus P(-1.67 ≥ z ≥ 1.00)

=1 - .7938 = .2062

z