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Lesson 5: Continuous Random Variables
§5.1 Probability Density Function (pdf)
Satya Mandal, KU
September 18, 2013
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
”As far as the laws of mathematics refer to reality, they are
not certain; and as far as they are certain, they do not refer to
reality.” - Albert Einstein
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Goals
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◮
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In Lesson 4, Discrete Random Variables were discussed.
In §5.1, Continuous Random Variables will be defined.
Mainly, probability density function will be discussed.
There is no homework on this section.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Continuous Random Variables
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A continuous random variable was defined as a random
variable X that can assume any value on an interval.
Examples: height, weight, volume, time.
The Probability distribution of a Discrete Random
Variable was given by the probability function p(x).
Probability distribution of a continuous random variable
defined and behaves in a different manner than a discrete
random variable.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Probability Density Function (pdf)
Let X be a continuous random variable on a sample space S.
The probability distribution of X is defined as follows:
◮ There is a function f (x), of real numbers x, associated to
X to be called the probability density function,
(abbreviated as ”pdf”) of X . The pdf f (x) of X must
satisfy the following two properties:
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f (x) is always non-negative (i.e f (x) ≥ 0 for all x).
The total area under the graph of y = f (x) and above
the x−axis is one.
For any two real numbers a ≤ b (also for a = −∞ and
b = ∞) the probability that X will be between a and b is
given by the area under the graph of y = f (x), above the
x−axis and between the vertical lines x = a and x = b.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Continued
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Geometrically, the condition f (x) ≥ 0 for all x means that
the graph of the function y = f (x) always lies on or
above the x-axis.
Notationally, the last statement is:
P(a ≤ X ≤ b) = the area of the region
under the graph of y = f (x), above x − axis,
between the vertical lines x = a and x = b.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Continued
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Further, the same area
= P(a ≤ X ≤ b) = P(a ≤ X < b)
= P(a < X ≤ b) = P(a < X < b).
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So, given a pdf f (x) of a random variable X , probability
computation boils down to computation of area under the
graph y = f (x).
Review Animation 5.1.1 for examples of pdf and
probability.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
A Contrast
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In contrast to discrete random variable, for any real
number a,
P(X = a) = 0
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This is because P(X = a) = P(a ≤ X ≤ a) =
(area of the line x = a, from x −axis upto the pdf ) = 0.
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
mean µ and standard deviation σ
Let X be a continuous random variable.
◮ A formal definition of mean µ and variance σ 2 , of a
continuous random variable X , involves some knowledge
of Calculus (integration). For this course, a formal
definition will not be necessary.
◮ As in the case of discrete random variables, the mean µ
of X the represents the average value of X .
◮ The standard deviation σ of X is a measure of variability
of X .
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Some well known continuous random variables
Among the most well known continuous random variables are:
◮ The Normal random variable will be discussed extensively.
Two parameters, µ and σ determine the pdf y = f (x) of
such a variable. (Review Animation 5.1.2)
◮ The T-random variable will be discussed in lesson 7.
(Review Animation 5.1.3)
◮ The Chi Square (χ2 ) random variable will be discussed in
lesson 7. (Review Animation 5.1.4)
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens
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Continuous Random Variable and PDF
Continued
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The pdf of the Uniform(0,1) is given by
1 if 0 ≤ x ≤ 1
f (x) =
0 otherwise
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Similarly, for any two numbers a < b the Uniform(a, b) is
given by
1
if 0 ≤ x ≤ 1
b−a
f (x) =
0
otherwise
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Review Animation 5.1.5
Also review, Animation 5.1.6 for exponential random
variable.
◮
Satya Mandal, KU
Lesson 5: Continuous Random Variables §5.1 Probability Dens