Binomial Random and Normal Random Variables

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Binomial Random and Normal Random Variables

• Lecture XI

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Bernoulli Random Variables

• The Bernoulli distribution characterizes the coin
  toss. Specifically, there are two events X0,1
  with X1 occurring with probability p. The
  probability distribution function PX can be
  written as

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Sum of Two Bernoulli Variables

• Next, we need to develop the probability of XY
  where both X and Y are identically distributed.
  If the two events are independent, the
  probability becomes

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Transforming the Two Bernoulli Scenario

• Now, this density function is only concerned with
  three outcomes ZXY0,1,2. There is only one
  way each for Z0 or Z2. Specifically for Z0,
  X0 and Y0. Similarly, for Z2, X1 and Y1.
  However, for Z1 either X1 and Y0 or X0 or
  Y1. Thus, we can derive

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Expanding to Three Bernoulli Variables

• Next we expand the distribution to three
  independent Bernoulli events where
  ZWXY0,1,2,3.

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Rewriting in Terms of Z

• Again, there is only one way for Z0 and Z3.
  However, there are now three ways for Z1 or Z2.
  Specifically, Z1 if W1, X1 or Y1. In
  addition, Z2 if W1 and X1, W1 and Y1, and
  X1 and Y1. Thus the general distribution
  function for Z can now be written as

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Binomial Distribution

• Based on this development, the binomial
  distribution can be generalized as the sum of n
  Bernoulli events.
• For the case above, n3. The distribution
  function (ignoring the constants) can be written
  as

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Explaining the Constants

• The next challenge is to explain the Crn term.
  To develop this consider the polynomial
  expression

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Pascals Triangle

• This sequence can be linked to our discussion of
  the Bernoulli system by letting ap and b(1-p).
  What is of primary interest is the sequence of
  constants. This sequence is usually referred to
  as Pascals Triangle

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1
1
1
1
1
2
1
3
3
1
1
1
4
6
4
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• This series of numbers can be written as the
  combinatorial of n and r, or
• Thus, any quadratic can be written as

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• As an aside, the quadratic form (a-b)n can be
  written as

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• Thus, the binomial distribution XB(n,p) is then
  written as

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Expected Value of Binomial

• Next recalling Theorem 4.1.6 EaXbYaEXbEY
  , the expectation of the binomial distribution
  function can be recovered from the Bernoulli
  distributions

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Variance of Binomial

• In addition, by Theorem 4.3.3
• Thus, variance of the binomial is simply the sum
  of the variances of the Bernoulli distributions
  or n times the variance of a single Bernoulli
  distribution

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