Continuous Probability Distributions Uniform Probability Distribution Area as a measure of...
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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