Transcript Chapter 8 Continuous Random Variables

Chapter 8
Continuous Random
Variables
Stat-Slide-Show, Copyright 1994-95 by Quant Systems Inc.
Prof. Vinod
Introduction
8- 2
Continuum
• The previous chapter was
primarily devoted to random
variables that were counts of
some phenomenon that was
called a “success.”
• These counts could only take on
discrete values usually starting at
zero.
• In this chapter the focus will be
on variables that take on any
value in some continuum, where
a continuum is simply a range of
numbers on the real number line.
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Example 1
• The range of newborn baby weights
lies on a continuum between 2 and
13 pounds.
[
]
2
13
pounds
• Observations measured in a
continuum can be very close
together; for example, two weights
could be
8.5672538764522134599... inches, and
8.5672538764522134598... inches.
8- 4
Discrete or Continuous?
• Many continuous variables are
discretized (rounded) by the
instruments used to measure
them.
• It is important to distinguish
the nature of the variable from
its physical measurement.
• Variables like heights and
weights are continuous
random variables even though
they are usually given to the
nearest unit.
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Continuous vs. Discrete
• One of the striking differences
in discrete and continuous
variables concerns the way in
which probability is defined.
• In a probability distribution for
a discrete random variable,
each possible outcome of the
random variable is assigned its
own probability.
• However, for continuous
random variables, there are
infinitely many outcomes, and
each has no probability.
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Outcomes of a Continuous
Random Variable
• Outcomes of a continuous
random variable do not have
probability assigned to any
one point, because there are
too many points.
• If we attempted to assign each
value, even an infinitesimal
probability, the sum of all the
probabilities would exceed
one. (sum >1?)
• Thus, for continuous random
variables, probability is only
assigned to intervals.
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Probability is
area,(strict=fuzzy)
• Since probability is assigned to
intervals when x is a continuous random variable (rv),
the probability is associated
with the area under the pdf
along the specified interval.
• Bottom line strict inequality
and the fuzzy (e.g.  ) are
equivalent for continuous rv:
• P(x<some constant)
= P(xthat same constant)
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The Continuous
Uniform Distribution
8- 9
Continuous Uniform
Distribution
• Both uniform distributions distribute
probability evenly across a sample
space.
• For the continuous uniform
distribution, the probability density
is spread out evenly over some
range a to b as shown in the figure
below.
1
b-a
a
b
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Probability Density
Functions (p.d.f)
• Continuous random variables
do not have probability
distribution functions.
• Instead, they have probability
density functions which are
denoted f(x).
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The Uniform Probability
Density Function
 1
,

f(x) =  b - a
0
for a  x  b
otherwise
The parameters of the density
function are the minimum and
maximum value of the random
variable and are referred to as
a and b, respectively.
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Expected Value and
Standard Deviation
• The expected value (mean) of
the continuous uniform
random variable is
ab
b-a

. and  =
2
12
• The standard deviation of the
continuous uniform random
variable is
ab
b -a

an  =
.
2
12
8-
Shape of the Continuous
Uniform Distribution
• When the uniform density function
is graphed, it produces a rectangle
or square.
• The probability of observing a
random variable in some interval is
expressed as an area under the
density function associated with the
interval.
1
b-a
a
b
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Calculating the
Probability of an
Interval
Because the density function for
the uniform distribution produces
a rectangle, calculating the
probability of an interval requires
very simple geometry.
Area of a rectangle
=Height * width
We want area to be 1, width
from a to b is b-a, hence the
height must be 1/(b-a)
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Example 2 (Time to
reach the fire)
• The fire department records how
long it takes each of its trucks to
reach the scene of a fire.
• Suppose that the distribution of
arrival times is uniform with the
minimum time being 2.0 minutes
and the maximum time being 15
minutes.
• What is the probability that a truck
will reach a reported fire scene
within five to ten minutes?
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Ex.2 – Solution
(5<time<10 minutes)
Area = Height

Width
1 =
1 = 1
Height = b - a 15 - 2 13
1
13
2
5
Area = 1
13
10

15
(10 - 5) = 5
13
P(5  X  10) = 5
13
8-
Ex.2continued
expected timeE(X), V(X)
• What is the expected time
until arrival?
E(X) = = a + b = 15 + 2 = 8.5 min.
2
2
• What is the standard deviation of
truck arrival times?
= b - a = 15 - 2 = 3 .753 min.
12
12
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What does data from a
uniform random variable
look like?
• While the density function for the
uniform distribution has a flat top,
a histogram of data from a
uniformly distributed random
process will not be so perfectly
flat.
100 observations
1000 observations
20
130
18
120
16
110
100
14
90
12
80
10
70
8
60
6
50
4
40
2
30
2
3.4
4.8
6.2
7.6
9
10.4
11.8
13.2
14.6
2
3.4
4.8
6.2
7.6
9
10.4
11.8
13.2
14.6
• When more observations are
generated, the top of the
distribution will begin to level.
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The Normal
Distribution
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Normal Distribution
• The normal distribution was
originally called the Gaussian
distribution, named after Karl
Gauss who published a work
in 1833 describing the
mathematical definition of the
distribution.
• Gauss developed this
distribution to describe
the error in predicting
the orbits of planets.
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Shape of the Normal
Distribution
• Normal distributions are all bell
shaped, but the bells come in
various shapes and sizes.
• However, since normal
distributions are all bell shaped
and symmetric, the mean,
mode, and median are equal.
mean
median
mode
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Properties of the
Normal Distribution
Although the distribution can
range in value from minus
infinity to positive infinity, red
values that are a great
distance from the mean rarely
occur. Range: -4<z<4
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Properties of the
Normal Distribution
One of the more important
properties of normal random
variables is that within a fixed
number of standard deviations
from the mean, all Normal
densities contain the same
fraction of their probabilities.
Remember one-sigma or twosigma rules? We now show
how they are related to area
under the standard Normal
density z~N(0,1) curve.
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Area Under the Curve
• In the figure, the shaded area
represents the probability of being
within  1 of the mean.
• The probability of a normal random
variable being in some interval
corresponds to the shaded area
under the curve.
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Area Under the Curve

• The total area under the curve from
- to  equals one.
• The shaded area under the curve
and the probability of being within
one standard deviation (1) of the
mean equals .68.
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Area Under the Curve
2
The shaded area in the figure
above represents the probability
of being within  2 of the mean
which equals .9475 for every
normal distribution.
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Area Under the Curve
3
• The shaded area in the figure
above represents the probability
of being within  3 of the mean
which equals .997 for every
normal distribution.
As you can see virtually all of the
area under the curve is within
three standard deviations of the
mean.
8-