# Hw8 - 4.1.6. (a) Since the interarrival times are...

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I I 4.1.6. (a) Since the interarrival times are independent and have the exp()) distribution, the joint c.d.f. and p.d.f. are the products of marginals: F(s1,s2): (1-e-r'r)(1-e-r'r;, .f(sr,sz) = ()e-)"')()u-)"r), sr,sz ) 0. (b) VrL T1*Tz de * ,eW{*# P[T, < 1, ?2 > 3] = P[St<1,Sr*Sz>3] 1@ lf : I 2"-"t I 2"-rt, ds2ds1 JJ 0 3-c1 I = | 2s-2""-z(e-"t) dsr J 0 : 2e-6 /5rrSt=3 7 = I 3e-,e-3vdu J- o @ r : e-" I 3e-sYdu t- 0 : - o-,o-3vlt6 t0 : e-o, The conditional density can be found by using the definition of f(ylx): f (ylr) = f fr,lJ _ 3e-(c+sv) :3e-Br/. ' I"\x) e-' The marginal density of Y, which is .found by integrating the joint den- sity over the interval [0, -] with respectto.r, ulso co*es out to 3e-ss. Therefore the random variables are independent. lr0 = i,"-,,*,u,0, u6 4.2.9. (a) We need to comPute W+huca;t, 1 I f(x,I/2,7,2) f@iY=;,2:i):ffi= for values of c in (0,1). 4.2.6. (MM) To find the conditional density of y given c , first integrate the joint density with respect to gr over the interval-[0, mJ in order to find the marginal density of a. l(*l;,{)

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## This note was uploaded on 10/27/2010 for the course MATH 351 taught by Professor Moumen,f during the Spring '08 term at George Mason.

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Hw8 - 4.1.6. (a) Since the interarrival times are...

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