Expected Value, Variance and Moment Generating Function

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Tutorial 4
Expected Value, Variance and Moment Generating
Function
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Expected Value

• If X is a discrete random variable having a
  probability mass function p(x), the expectation
  or the expected value of X, is defined by
• EX is a weighted average of the possible values
  that X can take on, each value being weighted by
  the prob. that X assumes it.

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Expected Value

• If X is a continuous random variable having a
  probability density function f(x), then the
  expected value of X is defined by
• If a and b are constants, then

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Example 1

• Calculate EX if X if a Poisson random variable
  with parameter .

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Example 2

• Calculate the expectation of a random variable
  uniformly distributed over (a, b).
• The expected value of a random variable uniformly
  distributed over the interval (a, b) is just the
  midpoint of the interval.

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Variance

• If X is a random variable with mean ,
• then the variance of X is defined by
• The variance of X measures the expected square of
  the deviation of X from its expected value.

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Variance
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Variance

• If a and b are constants, then
• The square root of the Var(X) is called the
  standard deviation of X.

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Example 3

• Calculate Var(X) when x represents the outcome
  when a fair die is rolled.

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Moment Generating Functions

• Moment generating function is define as follow

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Moment Generating Functions
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Moment Generating Functions

• Similarly,

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Moment Generating Functions

• In general, the nth derivative of M(t) is given
  by
• Implying that

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Example 4

• Binomial Dist. (n,p)

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Example 4 (cont)

• Mean M(0) np
• EX2 M(0) n(n-1)p² np
• Variance EX2 (EX)²
• n(n-1)p² np (np)²
• np(1-p)