MA523 Dr. Imad Khamis

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MA523Dr. Imad Khamis

• Chapter 4. Bivariate Distributions
• Fall 2006

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Two And Higher Dimensional R. V.'s

• In many statistical investigations, one is
  frequently interested in studying the
  relationship between two or more r.v.'s, such as
  the relationship between annual income and yearly
  savings per family or the relationship between
  occupation and hypertension. In this chapter, we
  consider n-dimensional vector valued r.v.'s,
  however, we start with the case of n 2.

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• Let E be a statistical experiment with a sample
  space S. Let X(s) and Y(s) be two functions each
  assigning a real number to each outcome s?S by
• (X,Y)(s) (X(s),Y(s)).
• We call the pair (X, Y) a two-dimensional random
  vector or a bivariate r.v. The joint cumulative
  distribution function of (X,Y) is defined by,
• F(x, y) P(X ? x, Y ? y),?(x, y) .

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• Definition (X, Y) is called a two-dimensional
  discrete r.v., if the possible values of (X, Y)
  are finite or countable, i.e.
• R(X, Y) (x1, y1), (x2, y2), ....
• Similarly, (X, Y) is said to be continuous
  two-dimensional r.v.'s if (X, Y) assumes a
  continuum of values in some sub-set of the
  Euclidian plane R2, i.e. R(X, Y) (x, y) (x,
  y) A where A ? R2

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Definition

• Let (X, Y) be a 2-dim. Discrete r.v. with each
  value of (xi , yi), the probability
• P(X xi ,Y yj ) p(xi , yj )
• is called the joint probability (mass)
  function (jpmf), and it satisfies the conditions

Definition The joint cumulative distribution
function (jcdf ) is given by
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• Definition
• A bivariate r.v. (X, Y) is called continuous
  r.v. if its jpdf F is twice differentiable such
  that
• f(., .) is called the joint probability density
  function (jpdf) of (X, Y) and satisfies the
  following conditions
• f(x, y) ? 0, (x, y) ? R2,
• P(X ? x, Y ? y) F(x, y).

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Independence of R.V.'s

• Definition Two r.v.'s X and Y are called
• independent, if
• P(X?A, Y?B) P(X?A) P(Y?B), ? A B.
• ? F(x, y) FX(x) FY(y) , ?x y
• p(x,y) p(Xx) p(Yy) ?x y discrete case,
• ? f(x,y) fX(x) fY(y) ?x y continuous case.

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• Example p(x, y) (x2 y)/7, (x,y)(1,1),(1,2),(2,1
  ) 0 o/w
• p(1,1) 1/7, pX(1) 1/7 2/7 3/7, pY(1)
  5/7
• Since p(1,1) ? p(X1) p(Y1), X Y are not
  independent (or X Y are dependent.
• Example)
• (X,Y) f(x,y) x2 e -x(y1), if x gt0 y gt 0 0
  o/w

Since f(x,y) ? fX(x) fY(y),? x y,?X and Y are
not independent.
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• Marginal and Conditional Distributions
• Discrete case Notice that
• P(X xi) pX(xi) P(X xi,Y y1,Y y2, )
• Similarly,
• pY(yj) P(Y yj)
• pX(xi) and pY(yj) are called the marginal pmf of
  X
• and Y respectively.

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Exercise P(X gt Y) p(1, 0) p(2, 0) p(2, 1)
2/5 P(X Y?2) p(1, 0) p(1, 1) p(2, 0)
7/25 Ex. 5. For 0? x ?4, 0? y ?4, 0? x y ?4 ,
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Continuous case

• Similarly, in the continuous case, we define the
  marginal pdf's of X Y. Let X and Y be two
  continuous r.v.'s with jpdf f(x, y). The
  marginal pdf 's of X and Y are defined by

Now, P(a ? X ? b, c ? Y ? d)
f(x, y) dx dy.
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• Problem
• Let (X, Y) f(x, y) x2 (xy)/3, 0 ? x ? 1, 0 ?
  y ?2,
• find P(X ? (1/2))
• 
  x2y (xy2)/6 2x2 (2x)/3.

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Conditional Distributions

• Definition
• Let (X, Y) be a discrete r.v. with jpmf, p(xi, yj
  ).
• The conditional pmfs are defined by
• p( xi?yj ) P(X xi ?Y yj) , pY(yj) ? 0

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• Definition The conditional pdf of X given that
  Yy is defined by
• for all y ? fY(y) ? 0.
• The conditional cdf of X given that Yy is
  defined by
• Therefore,

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Problem 10 p. 327 (X, Y) f(x, y) (1/2)
e-x, for x ? 0, ?y?lt x.
y
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• Conditional Expectation
• The conditional expectation of a r.v. X given
  that Yy is defined in the discrete case by
• and in the continuous case by

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Ex. continued

• .

Conditional Variance Furthermore, the conditional
variance of X given Y y is defined by V(X?Y
y) E(X2 Y y) - E(X?Y y) 2.

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Covariance joint probability

• The covariance measures the strength of the
  linear relationship between two variables
• The covariance

where X discrete variable X Xi the ith
outcome of X Y discrete variable Y Yi the
ith outcome of Y P(XiYi) probability of
occurrence of the condition affecting the ith
outcome of X and the ith outcome of Y
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The Sample Covariance

• The sample covariance

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Interpreting Covariance

• Covariance between two random variables
• cov(X,Y) gt 0 X and Y tend to move in the
  same direction
• cov(X,Y) lt 0 X and Y tend to move in
  opposite directions
• cov(X,Y) 0 X and Y are independent