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)r^3-Gb'a=E]suc\`?l?KPZ>AD6,"K,.Q2C]m^!sn],^?iUc8o+Q/Dl'! ;AbRZ.cCK([/`>3J") Given pdf of $X$, find a function $U$ that has the same distribution as $X$ where $U\sim Unif (0,1)$, Given X that is unif(0,1), find the pdf of Y=1/sqrt(X), Fnd the pdf of $\cos(X)$ when $X$ has pdf $f(x)=\frac{x}{2\pi^2}$ for $0 f conditional = f joint Pr conditional = 1 1 2 1 3 = 6. where the 1 on top is the usual uniform-over-square, the 1 / 2 below is for X < Y, and the 1 / 3 comes from the fact "It's not hard to show that they all have probability 1 / 3 of being the shortest." Allow me to be pedantic. 9>kM@f4t^?P_5:lFFX+Bksg%KZJ@JQX7t3mY/0Zu=\h.TL4^ :r^MOPg(s4`9PL&Hi1rdt\JOCjT>irV b:#e8V["pEJUZGNoTn@LL26FI*`S7r8%R!O]]f:=1&O(YCR\`[9Rl$r7tQ;sW0%fM F,K"V(:DD'j;JpRiFcQ"YD:h8rn\h2QQrk\AM?.%O4=(ML_AJJkV+Z3nbA;9"67*% V?0IAKal0"s)FtSVKHbZ%e@_Zfi19gkBEX([GT..9XPn>uq$L (=!iV4D_f'FI\5DA'M"RlZ5!oSB]gH--F*u\?ecgG<25Yn-0P/CLoB^EZj.0<0-[h8^-=> dEL+g[TYIs+[P[KbjCtYP,3__"dq'V=G;K-o?Edc4iGe! Let Z = min {X1,X2,X3}. }$$, If we assume that the derivative of $u^{-1}$ is always positive, setting $h = u^{-1}$, we have the differential equation 4f+/]$L*KA9@"2]-5b4Z/DFJJkM.Z"S9CrATdVV/hWb=\KaSg"/3J`iqVXh2B$W=fD!#^SLad*'L,c!p_'8=s(Pl a,m2"HnCUu!o_! 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4hgQA4bGl?4\%UfHGNq(?>? J*/Fd?%Z9qd@enQ%Aq.eX5\1<6ZiK4muI=ZNC3f?0A"S*C+RdX,S%(bkU05\-lB/)i&A/BK*TlYTo%T2bO%i] !%Ica>pNS&(lsDF%6i_l,bOet ,ch[8?u+Q$clu4g-4nAM5%$H2o5mlg2p^k'] D3&r]S4CL\S6/uTku:8cm7I]+jj);Eik1ce2ljgg!Z6,(ll7Yf$U l@:qU&TC;9JL/&2HgIX/HtWXhI8^c#nANM98FTDGZd9\SFo<1`(%$=! Wm"e2SRHJHc">%&hbpYVes%Tp(9o? 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