handout_week5_b - Stochastic Signals and Systems Multiple...

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Unformatted text preview: Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Example Find the normalization constant c and the marginal pdf’s for the following joint pdf: f X , Y ( x , y ) = ce- x e- y ≤ y ≤ x < ∞ elsewhere Problem Let X and Y denote the amplitude of noise signals at two antennas. The random vector ( X , Y ) has the joint pdf f X , Y ( x , y ) = axe- ax 2 / 2 bye- by 2 / 2 x > , y > , a > , b > a. Find the joint cdf. b. Find P [ X > Y ] . c. Find the marginal pdf’s. Independence of Random Variables Two rvs X and Y are called (statistically) independent if the events { X ∈ A } and { Y ∈ B } are independent, that is, if P [ X ∈ A , Y ∈ B ] = P [ X ∈ A ] P [ X ∈ B ] where A and B are two arbitrary sets on the x and y axes, respectively. Equivalently, X and Y are independent if and only if their joint cdf is equal to the product of their marginal cdfs, F X , Y ( x , y ) = F X ( x ) F Y ( y ) Similarly, X and Y are independent if and only if their joint pdf...
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handout_week5_b - Stochastic Signals and Systems Multiple...

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