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Unformatted text preview: rdθ.
0 0 Diﬀerentiating FR to obtain fR yields
2π fR (c) =
dc c fX,Y (r cos(θ), r sin(θ))rdr dθ
0 2π = fX,Y (c cos(θ), c sin(θ)) c dθ. (4.19) 0 The integral in (4.19) is just the path integral of fX,Y over the circle of radius c. This makes sense,
because the only way R can be close to c is if (X, Y ) is close to the circle, and so (4.19) is a
continuous-type example of the law of total probability. 144 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES A special case is when fX,Y is circularly symmetric, which by deﬁnition means that fX,Y (u, v )
depends on (u, v ) only through the value r = u2 + v 2 . Equivalently, circularly symmetric means
that fX,Y (u, v ) = fX,Y (r, 0). So, if fX,Y is circularly symmetric, (4.19) simpliﬁes to:
fR (c) = (2πc)fX,Y (c, 0) (if fX,Y is circularly symmetric). Example 4.6.4 Suppose W = max(X, Y ), where X and Y are independent, continuous-type
random variables. Express fW in terms of fX and fY . (Note: This example complements the
analysis of the minimum of two random variables, discussed in Section 3.9 in co...
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- Spring '08
- The Land