Download Chapter 6:Jointly Distributed Random Variables

Chapter 6:Jointly Distributed
Random Variables
Joint cumulative probability distribution function of X and Y
F (a, b) = P{ X ≤ a, Y ≤ b}.....,−∞ < a, b < ∞
If X and Y are jointly continuous if there exists a function f( x, y)
defined for all real x and y in set C,
P{( X , Y ) ∈ C} =
∫∫ f ( x, y )dxdy
( X ,Y )∈C
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If A and B are any sets of real numbers, then by defining
C = {(x, y) : x in A and y in B}
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Example:
P{ X ∈ A, Y ∈ B} = ∫B ∫A f ( x, y )dxdy
f ( x, y ) is the joint probability density function
F (a, b) = ∫−b∞ ∫−a∞ f ( x, y ) dxdy
F(a,b) is the cumulative distribution function
f x ( x) = ∫−∞∞ f ( x, y )dy
probability density function of X
f y ( y ) = ∫−∞∞ f ( x, y )dx
probability density function of Y
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Example 1c (on page 262)
The joint density function of X and Y is given by
Example 1e (on page 265)
The joint density of X and Y is given by
e − x e −2 y ,......, 0< x <∞ , 0< y <∞
f ( x, y ) =  02,...,
 otherwise
− ( x + y ) ......, 0< x <∞ , 0< y < ∞
f ( x, y ) =  e0,...,otherwise

Find the density function of the random variable X/Y
Find (a) P{X > 1, Y < 1}
(b) P{X < Y}
(c) P{X < a}
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Find the marginal probability distribution of X, for instance X = 3
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Marginal probability distribution functions for X and Y
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Determine the conditional probability density function for Y given X =x
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Independent Random Variables Application
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A common measure of the relationship between two random
variables is the covariance. To define the covariance, we need to
describe the expected value of a function of two random variables
h(X,Y ).
That is, E [h (X, Y)] can be thought of as the weighted average
of h(x, y) for each point in the range of (X,Y).
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Joint Distribution of X and Y
Joint Distribution of X and Y
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X and Y
are
independent
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