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Chapt 6b - 6.4 Sums of Independent Random Variables 207...

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6.4 Sums of Independent Random Variables 207 Figure 6.8 PDF of Z = X + Y Assume it has not snowed up until now. What is the PDF of the time U until the second snowstorm? Solution Let X be the random variable that denotes the time until the first snowstorm from the reference time, and let Y be the random variable that de- notes the time between the first snowstorm and the second snowstorm. If we assume that the times between snowstorms are independent, then X and Y are independent and identically distributed random variables. That is, the PDF of Y is given by f Y ( y ) = braceleftbigg λ e λ y y 0 0 otherwise Thus, U = X + Y , and the PDF of U is given by f U ( u ) = integraldisplay 0 f X ( x ) f Y ( u x ) dx Since f X ( x ) = 0 when x < 0, f Y ( u x ) = 0 when u x < 0 (or x > u ). Thus, the range of interest in the integration is 0 x u , and we obtain f U ( u ) = integraldisplay u 0 f X ( x ) f Y ( u x ) dx = integraldisplay u 0 λ e λ x λ e λ( u x ) dx = λ 2 e λ u integraldisplay u 0 dx = λ 2 ue λ u u 0 trianglesolid This is the Erlang-2 distribution.
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208 Chapter 6 Functions of Random Variables Example 6.6 Find the PDF of U , which is the sum of X and Y that are indepen- dent random variables with the following PDFs: f X ( x ) = λ e λ x x 0 f Y ( y ) = λ 2 ye λ y y 0 Solution Since X and Y are independent and based on the argument developed in the previous example, the PDF of U is given by f U ( u ) = integraldisplay u 0 f X ( x ) f Y ( u x ) dx = integraldisplay u 0 λ e λ x λ 2 ( u x ) e λ( u x ) dx = λ 3 e λ u integraldisplay u 0 ( u x ) dx = λ 3 e λ u bracketleftbigg ux x 2 2 bracketrightbigg u 0 = λ 3 e λ u bracketleftbigg u 2 u 2 2 bracketrightbigg = λ 3 u 2 e λ u 2 = λ 3 u 2 e λ u 2 ! u 0 trianglesolid This is the Erlang-3 distribution. Example 6.7 Find the PDF of W , which is the sum of X and Y that are indepen- dent random variables with the following PDFs: f X ( x ) = λ e λ x x 0 f Y ( y ) = μ e λ y y 0 where λ negationslash= μ . Solution Since X and Y are independent, the PDF of W is given by f W ( w ) = integraldisplay −∞ f X ( x ) f Y ( w x ) dx The limits of integration can be derived as follows based on the facts presented in Figure 6.9. The lower limit is zero because f X ( x ) = 0 when x < 0. Also, f Y ( w x ) = 0 when w x < 0 (or when x > w ). Thus, the upper limit is w and the range of interest in the integration is 0 x w . trianglesolid
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6.4 Sums of Independent Random Variables 209 Figure 6.9 PDF of W = X + Y Thus, the PDF of W is given by f W ( w ) = integraldisplay w 0 f X ( x ) f Y ( w x ) dx = integraldisplay w 0 λ e λ x μ e μ ( w x ) dx = λ μ e μ w integraldisplay w 0 e μ ) x dx = λ μ λ μ e μ w braceleftbig 1 e μ ) x bracerightbig = λ μ λ μ braceleftbig e μ w e λ w bracerightbig w 0 Comment : The last three examples illustrate the fact that if f X ( x ) = 0 for x < 0 and f Y ( y ) = 0 for y < 0, then the PDF of the sum U = X + Y is given by f U ( u ) = integraldisplay u 0 f X ( x ) f Y ( u x ) dx = integraldisplay u 0 f X ( u y ) f Y ( y ) dy 6.4.1
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