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Unformatted text preview: 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 e y y otherwise Thus, U = X + Y , and the PDF of U is given by f U ( u ) = integraldisplay 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 f X ( x ) f Y ( u x ) dx = integraldisplay u e x e ( u x ) dx = 2 e u integraldisplay u dx = 2 ue u u trianglesolid This is the Erlang2 distribution. 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 ) = e x x f Y ( y ) = 2 ye y y 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 f X ( x ) f Y ( u x ) dx = integraldisplay u e x 2 ( u x ) e ( u x ) dx = 3 e u integraldisplay u ( u x ) dx = 3 e u bracketleftbigg ux x 2 2 bracketrightbigg u = 3 e u bracketleftbigg u 2 u 2 2 bracketrightbigg = 3 u 2 e u 2 = 3 u 2 e u 2 ! u trianglesolid This is the Erlang3 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 ) = e x x f Y ( y ) = e y y where negationslash= . 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 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 f X ( x ) f Y ( w x ) dx = integraldisplay w e x e ( w x ) dx = e w integraldisplay w e ( ) x dx = e w braceleftbig 1 e ( ) x bracerightbig = braceleftbig e w e w bracerightbig...
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
 Fall '07
 Carlton

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