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=Example. Sequence of uniformly distributed random variables= | =Example. Sequence of uniformly distributed random variables= | ||
− | Let <math>\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}</math> be <math>n</math> i.i.d. jointly distributed random variables, each uniformly distributed on the interval <math>\left[0,1\right]</math> . Define the new random variables <math>\mathbf{W}=\max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> | + | Let <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}</math> be <math class="inline">n</math> i.i.d. jointly distributed random variables, each uniformly distributed on the interval <math class="inline">\left[0,1\right]</math> . Define the new random variables <math class="inline">\mathbf{W}=\max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> |
and | and | ||
− | <math>\mathbf{Z}=\min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> . | + | <math class="inline">\mathbf{Z}=\min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> . |
− | (a) Find the pdf of <math>\mathbf{W}</math> | + | (a) Find the pdf of <math class="inline">\mathbf{W}</math> |
− | <math>F_{\mathbf{W}}(w)=P\left(\left\{ \mathbf{W}\leq w\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq w\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \cap\left\{ \mathbf{X}_{2}\leq w\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq w\right\} \right)</math><math>=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq w\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq w\right\} \right)=\left(F_{\mathbf{X}}\left(w\right)\right)^{n}</math> . | + | <math class="inline">F_{\mathbf{W}}(w)=P\left(\left\{ \mathbf{W}\leq w\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq w\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \cap\left\{ \mathbf{X}_{2}\leq w\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq w\right\} \right)</math><math class="inline">=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq w\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq w\right\} \right)=\left(F_{\mathbf{X}}\left(w\right)\right)^{n}</math> . |
− | where <math>f_{\mathbf{X}}(x)=\mathbf{1}_{\left[0,1\right]}(x)</math> and <math>F_{X}\left(x\right)=\left\{ \begin{array}{ll} | + | where <math class="inline">f_{\mathbf{X}}(x)=\mathbf{1}_{\left[0,1\right]}(x)</math> and <math class="inline">F_{X}\left(x\right)=\left\{ \begin{array}{ll} |
0 ,x<0\\ | 0 ,x<0\\ | ||
x ,0\leq x<1\\ | x ,0\leq x<1\\ | ||
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\end{array}\right.</math> . | \end{array}\right.</math> . | ||
− | <math>f_{\mathbf{W}}\left(w\right)=\frac{dF_{\mathbf{W}}\left(w\right)}{dw}=n\left[F_{\mathbf{X}}\left(w\right)\right]^{n-1}\cdot f_{\mathbf{X}}\left(w\right)=n\cdot w^{n-1}\cdot\mathbf{1}_{\left[0,1\right]}(w).</math> | + | <math class="inline">f_{\mathbf{W}}\left(w\right)=\frac{dF_{\mathbf{W}}\left(w\right)}{dw}=n\left[F_{\mathbf{X}}\left(w\right)\right]^{n-1}\cdot f_{\mathbf{X}}\left(w\right)=n\cdot w^{n-1}\cdot\mathbf{1}_{\left[0,1\right]}(w).</math> |
− | (b) Find the pdf of <math>\mathbf{Z}</math> . | + | (b) Find the pdf of <math class="inline">\mathbf{Z}</math> . |
− | <math>F_{\mathbf{Z}}(z)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=1-P\left(\left\{ \mathbf{Z}>z\right\} \right)=1-P\left(\left\{ \min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} >z\right\} \right)</math><math>=1-P\left(\left\{ \mathbf{X}_{1}>z\right\} \cap\left\{ \mathbf{X}_{2}>z\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>z\right\} \right)=1-\left(1-F_{\mathbf{X}}(z)\right)^{n}.</math> | + | <math class="inline">F_{\mathbf{Z}}(z)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=1-P\left(\left\{ \mathbf{Z}>z\right\} \right)=1-P\left(\left\{ \min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} >z\right\} \right)</math><math class="inline">=1-P\left(\left\{ \mathbf{X}_{1}>z\right\} \cap\left\{ \mathbf{X}_{2}>z\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>z\right\} \right)=1-\left(1-F_{\mathbf{X}}(z)\right)^{n}.</math> |
− | <math>f_{\mathbf{Z}}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=n\left(1-F_{\mathbf{X}}(z)\right)^{n-1}f_{\mathbf{X}}(z)=n\left(1-z\right)^{n-1}\mathbf{1}_{\left[0,1\right]}\left(z\right).</math> | + | <math class="inline">f_{\mathbf{Z}}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=n\left(1-F_{\mathbf{X}}(z)\right)^{n-1}f_{\mathbf{X}}(z)=n\left(1-z\right)^{n-1}\mathbf{1}_{\left[0,1\right]}\left(z\right).</math> |
− | (c) Find the mean of <math>\mathbf{W}</math> . | + | (c) Find the mean of <math class="inline">\mathbf{W}</math> . |
− | <math>E\left[\mathbf{W}\right]=\int_{-\infty}^{\infty}wf_{\mathbf{w}}(w)dw=\int_{0}^{1}nw^{n}dw=\frac{n}{n+1}w^{n+1}|_{0}^{1}=\frac{n}{n+1}.</math> | + | <math class="inline">E\left[\mathbf{W}\right]=\int_{-\infty}^{\infty}wf_{\mathbf{w}}(w)dw=\int_{0}^{1}nw^{n}dw=\frac{n}{n+1}w^{n+1}|_{0}^{1}=\frac{n}{n+1}.</math> |
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Latest revision as of 07:14, 1 December 2010
Example. Sequence of uniformly distributed random variables
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be $ n $ i.i.d. jointly distributed random variables, each uniformly distributed on the interval $ \left[0,1\right] $ . Define the new random variables $ \mathbf{W}=\max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $
and
$ \mathbf{Z}=\min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $ .
(a) Find the pdf of $ \mathbf{W} $
$ F_{\mathbf{W}}(w)=P\left(\left\{ \mathbf{W}\leq w\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq w\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \cap\left\{ \mathbf{X}_{2}\leq w\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq w\right\} \right) $$ =P\left(\left\{ \mathbf{X}_{1}\leq w\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq w\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq w\right\} \right)=\left(F_{\mathbf{X}}\left(w\right)\right)^{n} $ .
where $ f_{\mathbf{X}}(x)=\mathbf{1}_{\left[0,1\right]}(x) $ and $ F_{X}\left(x\right)=\left\{ \begin{array}{ll} 0 ,x<0\\ x ,0\leq x<1\\ 1 ,x\geq1 \end{array}\right. $ .
$ f_{\mathbf{W}}\left(w\right)=\frac{dF_{\mathbf{W}}\left(w\right)}{dw}=n\left[F_{\mathbf{X}}\left(w\right)\right]^{n-1}\cdot f_{\mathbf{X}}\left(w\right)=n\cdot w^{n-1}\cdot\mathbf{1}_{\left[0,1\right]}(w). $
(b) Find the pdf of $ \mathbf{Z} $ .
$ F_{\mathbf{Z}}(z)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=1-P\left(\left\{ \mathbf{Z}>z\right\} \right)=1-P\left(\left\{ \min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} >z\right\} \right) $$ =1-P\left(\left\{ \mathbf{X}_{1}>z\right\} \cap\left\{ \mathbf{X}_{2}>z\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>z\right\} \right)=1-\left(1-F_{\mathbf{X}}(z)\right)^{n}. $
$ f_{\mathbf{Z}}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=n\left(1-F_{\mathbf{X}}(z)\right)^{n-1}f_{\mathbf{X}}(z)=n\left(1-z\right)^{n-1}\mathbf{1}_{\left[0,1\right]}\left(z\right). $
(c) Find the mean of $ \mathbf{W} $ .
$ E\left[\mathbf{W}\right]=\int_{-\infty}^{\infty}wf_{\mathbf{w}}(w)dw=\int_{0}^{1}nw^{n}dw=\frac{n}{n+1}w^{n+1}|_{0}^{1}=\frac{n}{n+1}. $