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=Example. Sequence of exponentially distributed random variables= | =Example. Sequence of exponentially distributed random variables= | ||
| − | Let <math>\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}</math> be a collection of i.i.d. exponentially distributed random variables, each having mean <math>\mu</math> . Define | + | Let <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}</math> be a collection of i.i.d. exponentially distributed random variables, each having mean <math class="inline">\mu</math> . Define |
| − | <math>\mathbf{Y}=\max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> | + | <math class="inline">\mathbf{Y}=\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{Y}</math> . | + | (a) Find the pdf of <math class="inline">\mathbf{Y}</math> . |
| − | <math>F_{\mathbf{Y}}\left(y\right)=P\left(\left\{ \mathbf{Y}\leq y\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq y\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \cap\left\{ \mathbf{X}_{2}\leq y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq y\right\} \right)</math><math>=P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq y\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq y\right\} \right)=\left(F_{\mathbf{X}}\left(y\right)\right)^{n}=\left(1-e^{-y/\mu}\right)^{n}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> | + | <math class="inline">F_{\mathbf{Y}}\left(y\right)=P\left(\left\{ \mathbf{Y}\leq y\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq y\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \cap\left\{ \mathbf{X}_{2}\leq y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq y\right\} \right)</math><math class="inline">=P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq y\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq y\right\} \right)=\left(F_{\mathbf{X}}\left(y\right)\right)^{n}=\left(1-e^{-y/\mu}\right)^{n}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> |
| − | <math>f_{\mathbf{Y}}(y)=\frac{dF_{\mathbf{Y}}(y)}{dy}=n\left(1-e^{-y/\mu}\right)^{n-1}\cdot\frac{1}{\mu}e^{-y/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\frac{n}{\mu}e^{-y/\mu}\left(1-e^{-y/\mu}\right)^{n-1}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> | + | <math class="inline">f_{\mathbf{Y}}(y)=\frac{dF_{\mathbf{Y}}(y)}{dy}=n\left(1-e^{-y/\mu}\right)^{n-1}\cdot\frac{1}{\mu}e^{-y/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\frac{n}{\mu}e^{-y/\mu}\left(1-e^{-y/\mu}\right)^{n-1}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</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>=\left[1-\left(1-\left(1-e^{-z/\mu}\right)\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math><math>=\left[1-\left(e^{-z/\mu}\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\left(1-e^{-nz/\mu}\right)\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</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 class="inline">=\left[1-\left(1-\left(1-e^{-z/\mu}\right)\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math><math class="inline">=\left[1-\left(e^{-z/\mu}\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\left(1-e^{-nz/\mu}\right)\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> |
| − | <math>f_{Z}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=\frac{n}{\mu}e^{-nz/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> | + | <math class="inline">f_{Z}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=\frac{n}{\mu}e^{-nz/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)</math> |
| − | (c) In words, give as complete a description of the random variable <math>\mathbf{Z}</math> as you can. | + | (c) In words, give as complete a description of the random variable <math class="inline">\mathbf{Z}</math> as you can. |
| − | <math>\mathbf{Z}</math> is an exponetially distributed random variable with mean <math>\frac{\mu}{n}</math> . | + | <math class="inline">\mathbf{Z}</math> is an exponetially distributed random variable with mean <math class="inline">\frac{\mu}{n}</math> . |
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Latest revision as of 07:13, 1 December 2010
Example. Sequence of exponentially distributed random variables
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be a collection of i.i.d. exponentially distributed random variables, each having mean $ \mu $ . Define
$ \mathbf{Y}=\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{Y} $ .
$ F_{\mathbf{Y}}\left(y\right)=P\left(\left\{ \mathbf{Y}\leq y\right\} \right)=P\left(\left\{ \max\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} \leq y\right\} \right)=P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \cap\left\{ \mathbf{X}_{2}\leq y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}\leq y\right\} \right) $$ =P\left(\left\{ \mathbf{X}_{1}\leq y\right\} \right)P\left(\left\{ \mathbf{X}_{2}\leq y\right\} \right)\cdots P\left(\left\{ \mathbf{X}_{n}\leq y\right\} \right)=\left(F_{\mathbf{X}}\left(y\right)\right)^{n}=\left(1-e^{-y/\mu}\right)^{n}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right) $
$ f_{\mathbf{Y}}(y)=\frac{dF_{\mathbf{Y}}(y)}{dy}=n\left(1-e^{-y/\mu}\right)^{n-1}\cdot\frac{1}{\mu}e^{-y/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\frac{n}{\mu}e^{-y/\mu}\left(1-e^{-y/\mu}\right)^{n-1}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right) $
(b) Find the pdf of $ \mathbf{Z} $
$ F_{\mathbf{Z}}(z)=P\left(\left\{ \mathbf{Z}\leq z\right\} \right)=1-P\left(\left\{ \mathbf{Z}>z\right\} \right)=1-P\left(\left\{ \min\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} >z\right\} \right) $$ =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} $$ =\left[1-\left(1-\left(1-e^{-z/\mu}\right)\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right) $$ =\left[1-\left(e^{-z/\mu}\right)^{n}\right]\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right)=\left(1-e^{-nz/\mu}\right)\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right) $
$ f_{Z}(z)=\frac{dF_{\mathbf{Z}}(z)}{dz}=\frac{n}{\mu}e^{-nz/\mu}\cdot\mathbf{1}_{\left[0,\infty\right)}\left(y\right) $
(c) In words, give as complete a description of the random variable $ \mathbf{Z} $ as you can.
$ \mathbf{Z} $ is an exponetially distributed random variable with mean $ \frac{\mu}{n} $ .
