# Hw10 - 5/F/C t4 OR"cfw_lDie S v 5.2.L The joint density of...

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? 5/F/C t4, 5.1.3. (a) For y e [t,9], ,1,( P[r S v] + fv(s) = Ftr(a) (b) For s e [0, t], PIY Svl = + fv(y): Fi(0 : 6.L.4. (MM) In order to simulate values for the density given in the pic- ture, the c.d.f. must be found. The p.d-f. ca.n be found to be r/-\ [ r, oSz<1 t\z):\r'_r, 1ic<2 Since it is defined piecewise, the c.d.f. can be computed as the definite integral ofthe pieces. if 0Sr< 1;1= t't)ll Afber plotting the c.d.f., a value [/ is picked randomly from the vertical a>ris, between 0 a^nd L. Then from this point a horizontal line is fouowed to the graph of tbe c.d.f. Flom this intersection point a perpendicular line is dropped and the value on the horizontal axis is the sirnulated value. So X is the inverse inage F-l(U), which turns out tobe tN whentl <l/2 and 2 7r*ila7 g(u,sz) = ls2l - fi"'uit*uill', ur,vz € (-oo, oo)' I The mareinal density of Yr is TW efrL W th yaT gr(yr) = t ./J -f r^. -fi"-co?n?*ah/z 4r, *yrry/ -*- : L Ior"-,7<v1+t)/24r, rJ "'- 0 l. - 1 o-t d,u. = i'flVto c The third line results from the substitution u = u?(g?

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