Product Sets and Marginal Probability

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Chapter 5
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Product Sets and Marginal Probability

• Definition Cartesian Product
• B?C (x,y) x ?B, y ? C
• Since
• X ?B, Y? C ? ? ? (X(?),Y(?) ? B?C, we have
• P(X ?B, Y? C) P(X(?),Y(?) ? B?C). Therefore,
• P(X ?B) P(X ?B??)
• P(X ?B?Y?R)
• P(X ?B, Y?R)
• P((X,Y) ? B?R)

• Problem 2. P(altX?b, cltY?d)
• FXY(b,d) ? FXY(a,d) ? FXY(b,c)
• FXY(a,c)
• Proof Let W (??,a ?(??,d, and X (??,b
  ?(??,c, Y (a,b ?(c,d, and Z (??,b
  ?(??,d. It is easy to show (W?X) ?YZ. Also,
  (W?X)?Y?. Thus, P(W?X) P(Y) P(Z). Moreover,
  W?X (??,a ?(??,c. Hence P(W?X)P(W)P(X)?P(W?X
  ).
• Thus, P(Y) P(Z)?P(W) ?P(X) P(W?X).

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Joint and Marginal CDF

• Definition Joint CDF of X Y
• If X, and Y are independent,
• FXY(x,y) FX(x)FY(y)
• Obtain Marginal CDF from Joing CDF

• Example 5.1
• Hence,

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Joint Continuous Random Variables

• DefinitionTwo random variables X and Y are
  Jointly continuous if
• Example 5.3 Show
• is a valid joint continuous density function.
• Answer Show

• CDF of jointly continuous R.V.s
• Example 5.4

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Marginal Densities

• Definition Marginal Densities
• Example 5.5 Continue from example 5.4, find
  fX(x), fY(y).

• Alternatively, using example 5.1, we have

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Independence, Expectations, Conditional
Probability, etc.

• X and Y are independent iff
• FXY(x,y) FX(x) FY(y)
• fXY(x,y) fX(x) fY(y)
• Expectations
• Bi-variate Characteristic Func.
• Conditional joint density

• If X, Y are independent, then
• fYX(yx) fY(y)
• Define Conditional Expectation
• Then
• Also, the substitution law holds

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Examples

• Example 5.6 If XN(0,1), and fYX(yx)N(x,1),
  find fXY(x,y).
• Solution
• Example 5.7 If X exp(1), and fYX(yx) N(0,
  x2), find EY2 and EY2X3

• Solution Use conditional expectation
• Since Xexp(1), EX2 2.
• It can be shown, EX5 5! 120.

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Example 5.8

• Let X, Y be both continuous random variables. If
  Z X Y, find fZ(z).
• Solution

Hence,
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Example 5.9

• X and Y are both continuous random variables. If
  Z XY, find fZ(z) and FZ(z), given fXY(x,y).

• Then,

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Bivariate Normal Distribution

• In the most general form, the pdf of a bivariate
  normal distribution is shown above.

• ? Correlation coefficient
• X, Y are independent if ? 0

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More Bivariate Normal Distributions
Created using binormal.m
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Examples

• Example 5.10 U, V standard binormal
  distribution. Show that
• EUV ?.
• Solution Define
• then

• Example 5.11 U, V ??(u,v), find fVU and fUV.
• Solution
• By symmetry,