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| -rw-r--r-- | houdre/hw5ii.tex | 45 |
1 files changed, 42 insertions, 3 deletions
diff --git a/houdre/hw5ii.tex b/houdre/hw5ii.tex index 3e99232..51b3208 100644 --- a/houdre/hw5ii.tex +++ b/houdre/hw5ii.tex @@ -23,16 +23,55 @@ \def\q{\afterassignment\qq\qnum=} \def\qq{\qqq{\number\qnum}} \def\qqq#1{\bigskip\goodbreak\noindent{\bf#1)}\smallskip} -\def\fr#1#2{{#1\over #2}} -\def\var{\mathop{\rm var}\nolimits} -\def\infint{\int_{-\infty}^\infty} +\def\align#1{\vcenter{\halign{$##\hfil$&&$\hfil##$\cr#1}}} +\tabskip=1em \q7 +By continuity, $X$ has a density function, so +$$\E(X) = \int_0^\infty xf_X(x) dx = +x(F_X(x)-1)\big|_0^\infty - \int_0^\infty (F_X(x)-1) dx,$$ +by integration by parts. +Because $xF_X(x)|_0^\infty = 0,$ (since $\Pr(X\leq\infty)-1 = 0.$) +$$\E(X) = \int_0^\infty 1-F_X(x)dx.$$ + \q9 +$$F_{X'}(x) = \bigg\{\align{ + F_X(x)&\hbox{if }x<a,\cr + 1&\hbox{if }x\geq a.\cr}$$ + +For $x\leq a,$ the distribution function is the same because the +$X' \leq x$ when $X\leq x.$ For $x\geq a,$ $\Pr(X'\leq x) += \Pr(X<a) + \Pr(X\geq a) = 1.$ \q10 +$$f_X(x) = \bigg\{\align{ + e^{-x}&\hbox{if }x>0,\cr + 0&\hbox{if }x\leq0.\cr}$$ +$$Y = (X-2)/(X+1) \to YX+Y = X-2 \to (Y-1)X = -(2+Y) +\to X = -(2+Y)/(Y-1).$$ +$$dX/dY = -{3\over(Y-1)^2}.$$ +By 5.52, since $g(x)$ is strictly decreasing, +$$\align{f_Y(y)&=&-f_X\left(-{2+y\over y-1}\right) + \left(-{3\over(y-1)^2}\right)\hfill\cr + &=&\bigg\{\align{ + e^{2+y\over y-1}{3\over(y-1)^2}&-2<y<1.\cr + 0&{\rm otherwise.}\cr}\hfill\cr}$$ +Note that this bound is true because ${2+y\over y-1}\in(0,\infty)\iff +y\in(-2,1).$ + \q14 +$$f_X(x) = \bigg\{\align{ + 1&x\in[0,1],\cr + 0&{\rm otherwise}.\cr}$$ +By theorem 5.50, with $Y = g(X) = {3X\over 1-X} \to 3X-Y(1-X) = 3X+YX-Y += 0 \to X = {Y\over 3+Y},$ (note that $g(x)$ is strictly decreasing on +$(0,\infty).$) +$$f_Y(y) = -f_X(g^{-1}(y))[g^{-1}]'(y) = +f_X\left({y\over 3+y}\right){3\over(3+y)^2} = \bigg\{\align{ + {3\over(3+y)^2}&y>0,\cr + \hfil 0&{\rm otherwise}.\cr}$$ + \bye |