Slide 4a.1

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Chapter 4 Part a The Expectation and Variance
of Distributions

• We will be discussing
• The Algebra of Expectation
• The Algebra of Variance
• The Normal Distribution
• (These topics are needed for Chapter 5)

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The Expectation of a Discrete Random Variable
Assume we have a random variable, ai. If ai can
take on only certain values 1, 2, , J, then
we have as the definition of Expectation of ai,
or E(ai)
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The Expectation of a Continuous Variable
Imagine we have a scalar, ai, that can take on
any possible values with probability f(ai), i. e.
f(ai)
ai
By definition, the expectation of that scalar is
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Three Rules for E()
The Expectation of a Constant is That
Constant E(c) c
The Expectation of a Sum is the Sum of the
Expectations E(a b) E(a) E(b)
In the Expectation of a Linear Combination, a
Constant Matrix Can Pass Through E() E(Da)
DE(a) E(a'F) E(a')F
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The Variance of a Random Variable
The Variance of the Random Vector a is Given by
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The Variance of a Mean Centered Vector
If E(a) 0, i. e. a is mean centered, we have
just V(a) E(aa')
NB just because E(a) 0 doesnt mean that
E(aa') 0!
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The Variance Matrix
For mean centered a, we have V(a) E(aa')
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Two Rules for V()
Adding a Constant Vector Does Not Change the
Variance V(a c) V(a)
The Variance of a Linear Combination is a
Quadratic Form V(Da) DV(a)D'
Hint The Above Theorem Will Figure Many Many
Times in What Is to Come!
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The Normal Density Function
Consider a random scalar x. Under the normal
distribution, the probability that x takes on
the value xa is given by the equation
1.0
Pr(x)
x
0
xa
?
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The Standard Normal Density
If ? 0 and ?2 1, so that z (x - ?) / ? the
expression simplifies to
Note very common notation
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The Normal Distribution Function
According to the normal distribution function,
the probability that the random scalar x is less
than or equal to some value xb is
1.0
Pr(x)
x
0
xb
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Other Notational Conventions
For our scalar x that is distributed according to
the normal distribution function, we say x
N(?, ?2).
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The Standardized Normal Distribution
If we set z (x - ?) / ? then
Again, note the notational convention and that
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The Normal Ogive
1.0
?(z)
0
zb