Continuous Probability Distributions

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Continuous Probability Distributions
f(x)

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

X
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Uniform Probability Distribution
NY
Chicago

• 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.

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Uniform Probability Distribution
With every one-minute interval being equally
likely, the random variable x is said to have a
uniform probability distribution
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Uniform Probability Distribution
For the flight-time random variable, a 120 and
b 140
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Uniform Probability Density Function for Flight
time
The shaded area indicates the probability the
flight will arrive in the interval between 120
and 140 minutes
120
140
125
130
135
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Basic Geometry
Remember when we multiply a line segment times a
line segment, we get an area
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Probability as an Area
Question What is the probability that arrival
time will be between 120 and 130 minutesthat is
10
120
140
125
130
135
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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.
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E(x) and Var(x) for the Uniform Continuous
Distribution
Applying these formulas to the example of flight
times of Chicago to NY, we have
Thus
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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.
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Normal Probability Distribution

• It has been used in a wide variety of
  applications

Heights of people
Scientific measurements
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Normal Probability Distribution

• It has been used in a wide variety of
  applications

Test scores
Amounts of rainfall
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The Normal Distribution
Where µ is the mean s is the standard
deviation ? 3.1459 e 2.71828
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Normal Probability Distribution

• Characteristics

The distribution is symmetric, and is
bell-shaped.
x
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Normal Probability Distribution

• Characteristics

The entire family of normal probability
distributions is defined by its mean m and its
standard deviation s .
Standard Deviation s
x
Mean m
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Normal Probability Distribution

• Characteristics

The highest point on the normal curve is at the
mean, which is also the median and mode.
x
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Normal Probability Distribution

• Characteristics

The mean can be any numerical value negative,
zero, or positive.
x
-10
0
20
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Normal Probability Distribution

• Characteristics

The standard deviation determines the width of
the curve larger values result in wider, flatter
curves.
s 15
s 25
x
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Normal Probability Distribution

• Characteristics

Probabilities for the normal random variable
are given by areas under the curve. The total
area under the curve is 1 (.5 to the left of the
mean and .5 to the right).
.5
.5
x
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The Standard Normal Distribution
The Standard Normal Distribution is a normal
distribution with the special properties that is
mean is zero and its standard deviation is one.
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Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
s 1
z
0
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Cumulative Probability
Probability that z 1 is the area under the
curve to the left of 1.

z
0
1
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What is P(z 1)?
To find out, use the Cumulative Probabilities
Table for the Standard Normal Distribution
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Exercise 1

• Answer
• .9931
• 1-.9931.0069

• What is P(z 2.46)?
• What is P(z 2.46)?

z
2.46
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Exercise 2

• Answer
• 1-.9015.0985
• .9015

• What is P(z -1.29)?
• What is P(z -1.29)?

Red-shaded area is equal to green- shaded area
Note that
-1.29
z
1.29
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
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Exercise 3
What is P(.00 z 1.00)?
P(.00 z 1.00).3413
0
1
z
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Exercise 4
What is P(-1.67 z 1.00)?
P(-1.67 z 1.00).7938
Thus P(-1.67 z 1.00) 1 - .7938 .2062
0
1
-1.67
z