{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

handout_week6_b

# handout_week6_b - over the region R z F Z z = Z Z R z f XY...

This preview shows pages 1–4. Sign up to view the full content.

Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 One Function of Two RVs Given two random variables X and Y and a function g ( x , y ) , we form a new random variable Z as Z = g ( X , Y ) Given the joint pdf f XY ( x , y ) , how does one obtain f Z ( z ) , the pdf of Z ?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
One Function of Two RVs Three methods: 1 By determining the probability F Z ( z ) = P [ Z z ] = P [ g ( X , Y ) z ] . 2 By fixing one of the two random variables, say Y , at a particular one of its possible values y , and then using f Z ( z ) = Z -∞ f Z ( z | y ) f Y ( y ) dy . 3 “Auxiliary” variable. No general statement can be made as to how to decide in a given problem which method is the simplest. In attacking a problem of this kind, it is wise first to sketch contours in the XY -plane along which the function g ( X , Y ) is constant. One Function of Two RVs: Method 1 Calculate the cumulative distribution function of Z = g ( X , Y ) by determining the probability F Z ( z ) = P [ Z z ] = P [ g ( X , Y ) z ] . The event g ( X , Y ) z is represented by a certain region in the XY -plane, which is bounded by the curve g ( x , y ) = z . Call that region R ( z ) . The probability is found by integrating the joint pdf f XY (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) over the region R ( z ) , F Z ( z ) = Z Z R ( z ) f XY ( x , y ) dxdy . One Function of Two RVs: Method 1 Let Z = X + Y . Determine the cdf and pdf f Z ( z ) . (Cont.): The pdf of f Z ( z ) of Z = g ( X , Y ) is obtained by differentiating this cdf with respect to z . Note: Leibnitz rule: d dz Z b ( z ) a ( z ) g ( x , z ) dx = db ( z ) dz g ( b ( z ) , z )-da ( z ) dz g ( a ( z ) , z )+ Z b ( z ) a ( z ) ∂ ∂ z g ( x , z ) dx One Function of Two RVs: Method 1 The random variables X and Y have joint density f X , Y ( x , y ) = ± e-y ≤ x ≤ y < ∞ otherwise Find the probability density of the sum Z = X + Y . One Function of Two RVs: Method 1 Let Z = max ( X , Y ) . Determine the cdf and pdf f Z ( z ) . One Function of Two RVs: Method 1 The random variables X and Y have the joint density f X , Y ( x , y ) = ± x / y ≤ x ≤ y ≤ 2 otherwise Find the density of the product Z = XY ....
View Full Document

{[ snackBarMessage ]}