Chapter 5 Joint Probability Distribution

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Chapter 5Joint Probability Distribution
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Introduction

• If X and Y are two random variables, the
  probability distribution that defines their
  simultaneous behavior is a Joint Probability
  Distribution.
• Examples
• Signal transmission X is high quality signals
  and Y low quality signals.
• Molding X is the length of one dimension of
  molded part, Y is the length of another
  dimension.
• THUS, we may be interested in expressing
  probabilities expressed in terms of X and Y,
  e.g., P(2.95 lt X lt 3.05 and 7.60 lt Y lt 7.8)

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Two discrete random variables

• Range of random variables (X,Y) is the set of
  points (x,y) in 2D space for which the
  probability that X x and Y y is positive.
• If X and Y are discrete random variables, the
  joint probability distribution of X and Y is a
  description of the set of points (x,y) in the
  range of (X,Y) along with the probability of each
  point.
• Sometimes referred to as Bivariate probability
  distribution, or Bivariate distribution.

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Example 5.1 Joint probability distribution for X
and Y
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Joint probability mass function

• The joint probability mass function of the
  discrete random variables X and Y, denoted as
  fXY(x,y) satisfies

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Marginal probability distributions

• Individual probability distribution of a random
  variable is referred to as its Marginal
  Probability Distribution.
• Marginal probability distribution of X can be
  determined from the joint probability
  distribution of X and other random variables.
• Marginal probability distribution of X is found
  by summing the probabilities in each column, for
  Y, summation is done in each row.

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Example 5-3 Marginal probability distribution
for X and Y
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Marginal probability distributions (Cont.)

• If X and Y are discrete random variables with
  joint probability mass function fXY(x,y), then
  the marginal probability mass function of X and Y
  are
• where Rx denotes the set of all points in the
  range of (X, Y) for which X x and Ry denotes
  the set of all points in the range of (X, Y) for
  which Y y

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Mean and Variance

• If the marginal probability distribution of X has
  the probability function f(x), then
• R Set of all points in the range of (X,Y).
• Example 5-4.

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Conditional probability

• Example 5-5.
• Given discrete random variables X and Y with
  joint probability mass function fXY(X,Y), the
  conditional probability mass function of Y given
  X x is
• fYx(y) fXY(x,y)/fX(x) for fX(x) gt 0

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Conditional probability (Cont.)

• Because a conditional probability mass function
  fYx(y) is a probability mass function for all y
  in Rx, the following properties are satisfied
• (1) fYx(y) ? 0
• (2) fYx(y) 1
• (3) P(YyXx) fYx(y)
• Example 5-6.

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Example 5-6 Conditional probability
distribution for Y given X
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Conditional probability (Cont.)

• Let Rx denote the set of all points in the range
  of (X,Y) for which Xx. The conditional mean of
  Y given X x, denoted as E(Yx) or ?Yx, is
• And the conditional variance of Y given X x,
  denoted as V(Yx) or ?2Yx is
• Example 5-7

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Independence

• Example 5-8
• For discrete random variables X and Y, if any one
  of the following properties is true, the others
  are also true, and X and Y are independent.
• (1) fXY(x,y) fX(x) fY(y) for all x and y
• (2) fYx(y) fY(y) for all x and y with fX(x) gt
  0
• (3) fXy(x) fX(x) for all x and y with fY(y) gt
  0
• (4) P(X ? A, Y ? B) P(X ? A)P(Y ? B) for any
  sets A and B in the range of X and Y
  respectively.
• If we find one pair of x and y in which the
  equality fails, X and Y are not independent.

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Joint and Marginal probability Conditional
probability distribution for X and
Y Distribution for X and Y
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Rectangular Range for (X, Y)

• If the set of points in two-dimensional space
  that receive positive probability under fXY (x,
  y) does not form a rectangle, X and Y are not
  independent because knowledge of X can restrict
  the range of values of Y that receive positive
  probability.
• Example 5-1
• If the set of points in two dimensional space
  that receives positive probability under fXY(x,
  y) forms a rectangle, independence is possible
  but not demonstrated. One of the conditions must
  still be verified.

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ANNOUNCEMENTS

• Assignment VII
• 5, 9, 10, 11, 12

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Two continuous random variables

• Analogous to the probability density function of
  a single continuous random variable, a Joint
  probability density function can be defined over
  two-dimensional space.

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Joint probability distribution

• A joint probability density function for the
  continuous random variables X and Y, denoted as
  fXY(x,y), satisfies the following properties
• (1) fXY(x,y) ? 0 for all x, y
• (2)
• (3) For any range R of two-dimensional space

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Joint probability distribution (Cont.)

• The probability that (X,Y) assumes a value in the
  region R equals the volume of the shaded region.

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Joint probability density function for the
lengths of different dimensions of an
injection-molded part P(2.95 lt X lt 3.05,7.60 lt
Y lt 7.80)
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Joint probability distribution (Cont.)

• Example 5-15

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Marginal probability distribution

• If the joint probability density function of
  continuous random variables X and Y is fXY(x,y),
  the marginal probability density function of X
  and Y are
• and
• where Rx denotes the set of all points in the
  range of (X,Y) for which X x and Ry denotes the
  set of all points in the range of (X,Y) for which
  Y y.

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Marginal probability distribution (Cont.)

• A probability involving only one random variable,
  e.g., P(a lt X lt b), can be found from the
  marginal probability of X or from the joint
  probability distribution of X and Y.
• For example
• P(a lt x lt b) P(a lt x lt b, - ? lt Y lt ?)

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Example 5-16
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Mean and variance

• E(x) ?x
• Where Rx denotes the set of all points in the
  range of (X,Y) for which Xx and Ry denotes the
  set of all points in the range of (X,Y)

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ANNOUNCEMENTS

• Assignment VII
• 5, 9, 10, 11, 12