From aa74b288fdb37b7d8947db0149ca31d4dfca4780 Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 10 Nov 2020 17:56:05 -0500 Subject: put down a bs answer for #20 that doesn't simplify anything really --- houdre/hw7.tex | 24 +++++++++++------------- 1 file changed, 11 insertions(+), 13 deletions(-) diff --git a/houdre/hw7.tex b/houdre/hw7.tex index 0d4a65e..7020669 100644 --- a/houdre/hw7.tex +++ b/houdre/hw7.tex @@ -33,16 +33,6 @@ \q2 -%Let the mgf of $X_i$ be $M_{X_i}(x).$ -%The sum of $n$ $X_i$ has mgf $M_{X_i}(x)^n.$ -%This has variance $(M_{X_i}(x)^n)''(0)-(M_{X_i}(x)^n)'(0)^2 = -%(nM_{X_i}(x)^{n-1}M_{X_i}'(x))'(0)-n^2\mu^2 = -%n(n-1)M_{X_i}(0)^{n-2}M_{X_i}'(0)^2 + nM_{X_i}(0)^{n-1}M_{X_i}''(0) -%-n^2\mu^2 = n(n-1)\mu^2 + n\sigma^2 - n^2\mu^2 = n\sigma^2 - n\mu^2.$ - -%The mgf of $\overline X$ is $$M_{\overline X}(x)=e^{tn^{-1}}M_{X_i}(x)^n. -%\var\overline X = (e^{tn^{-1}}(n^{-1}M_{X_i}(x)^n+nM_{X_i}(x)^{n-1}M_{X_i}'(x)^n))' $$ - $$\E\left(\fr1{n-1}\sum_{i=1}^n(X_i-\overline X)^2\right) = \E\left(\fr1{n-1}\sum_{i=1}^n(X_i-\mu - (\overline X - \mu))^2\right)$$$$ @@ -125,11 +115,19 @@ $$M_{A_n}(t) = \E(e^{itn^{-1}(X_1+\cdots+X_n)}) = By applying the partition theorem to $M(t) = \E(e^{tX}),$ we trivially get $$M(t) = \sum_{k=1}^\infty \Pr(N=k)\E(e^{tX}|N=k).$$ +Given that $\Pr(N=k) = {1\over(e-1)k!}$ and Given these specific distributions, $$\Pr(X\leq x) = \Pr(U_1\leq x)\cdots\Pr(U_k\leq x) = x^n \to -f_X(x) = nx^{n-1} \to \E(e^{tX}|N=k) = \sum_{j=0}^\infty {1\over j!}t^j -\E(n^jx^{j(n-1)}) = \sum_{j=0}^\infty {1\over -j!}t^jn^j{1\over1+j(n-1)}.$$ +f_X(x) = nx^{n-1}$$$$ \to \E(X^j) = \int_0^1 x^jnx^{n-1}dx = \int_0^1 +nx^{n+j}/(j+n)\to \E(e^{tX}|N=k) = \sum_{j=0}^\infty {1\over +j!}t^jnx^{n+j}/(j+n)$$$$ +\to M(t) = \sum_{k=1}^\infty {1\over(e-1)k!}\sum_{j=0}^\infty {1\over +j!}t^jnx^{n+j}/(j+n).$$ + using $\E(e^{tX}) = \sum_{n=0}^\infty {t^j\over j!}\E(X^j).$ +The difference of $R$ and $X$ must be exponentially distributed because +the quotient of their moment generating functions is similar to an +exponential moment generating function. + \bye -- cgit