Then let x r cos and y r sin 473 transformation

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Unformatted text preview: rdθ. 0 0 Differentiating FR to obtain fR yields 2π fR (c) = 0 d 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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