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Let <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}</math> be i.i.d. random variables with absolutely continuous probability distribution function <math class="inline">F\left(x\right)</math> . Let the random variable <math class="inline">\mathbf{Y}_{j}</math> be the <math class="inline">j</math> -th order statistic of the <math class="inline">\mathbf{X}_{i}</math> 's. that is: <math class="inline">\mathbf{Y}_{j}=j\text{-th smallest of }\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 i.i.d. random variables with absolutely continuous probability distribution function <math class="inline">F\left(x\right)</math> . Let the random variable <math class="inline">\mathbf{Y}_{j}</math> be the <math class="inline">j</math> -th order statistic of the <math class="inline">\mathbf{X}_{i}</math> 's. that is: <math class="inline">\mathbf{Y}_{j}=j\text{-th smallest of }\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\}</math> . | ||
| − | (a) | + | (a) What is another name for the first order statistic? |
| − | What is another name for the | + | (b) What is another name for the n/2 order statistic? |
| − | minimum | + | (c) Find the probability density function of the first order statistic. (You may assume n is odd.) |
| + | ---- | ||
| + | ==Share and discuss your solutions below.== | ||
| + | ---- | ||
| + | ==Solution 1== | ||
| + | (a) minimum | ||
(b) | (b) | ||
| − | |||
| − | |||
sample median | sample median | ||
(c) | (c) | ||
| − | |||
| − | |||
| − | |||
<math class="inline">F_{\mathbf{Y}_{1}}\left(y\right)=P\left(\left\{ \mathbf{Y}_{1}\leq y\right\} \right)=1-P\left(\left\{ \mathbf{Y}_{1}>y\right\} \right)</math><math class="inline">=1-P\left(\left\{ \mathbf{X}_{1}>y\right\} \cap\left\{ \mathbf{X}_{2}>y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>y\right\} \right)</math><math class="inline">=1-\prod_{i=1}^{n}P\left(\mathbf{X}_{i}>y\right)=1-\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n}.</math> | <math class="inline">F_{\mathbf{Y}_{1}}\left(y\right)=P\left(\left\{ \mathbf{Y}_{1}\leq y\right\} \right)=1-P\left(\left\{ \mathbf{Y}_{1}>y\right\} \right)</math><math class="inline">=1-P\left(\left\{ \mathbf{X}_{1}>y\right\} \cap\left\{ \mathbf{X}_{2}>y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>y\right\} \right)</math><math class="inline">=1-\prod_{i=1}^{n}P\left(\mathbf{X}_{i}>y\right)=1-\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n}.</math> | ||
<math class="inline">f_{\mathbf{Y}_{1}}\left(y\right)=\frac{d}{dy}F_{\mathbf{Y}_{1}}\left(y\right)=n\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n-1}f_{\mathbf{X}}\left(y\right).</math> | <math class="inline">f_{\mathbf{Y}_{1}}\left(y\right)=\frac{d}{dy}F_{\mathbf{Y}_{1}}\left(y\right)=n\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n-1}f_{\mathbf{X}}\left(y\right).</math> | ||
| + | ---- | ||
Latest revision as of 17:38, 13 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
January 2002
4. (20 pts)
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be i.i.d. random variables with absolutely continuous probability distribution function $ F\left(x\right) $ . Let the random variable $ \mathbf{Y}_{j} $ be the $ j $ -th order statistic of the $ \mathbf{X}_{i} $ 's. that is: $ \mathbf{Y}_{j}=j\text{-th smallest of }\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $ .
(a) What is another name for the first order statistic?
(b) What is another name for the n/2 order statistic?
(c) Find the probability density function of the first order statistic. (You may assume n is odd.)
Solution 1
(a) minimum
(b)
sample median
(c) $ F_{\mathbf{Y}_{1}}\left(y\right)=P\left(\left\{ \mathbf{Y}_{1}\leq y\right\} \right)=1-P\left(\left\{ \mathbf{Y}_{1}>y\right\} \right) $$ =1-P\left(\left\{ \mathbf{X}_{1}>y\right\} \cap\left\{ \mathbf{X}_{2}>y\right\} \cap\cdots\cap\left\{ \mathbf{X}_{n}>y\right\} \right) $$ =1-\prod_{i=1}^{n}P\left(\mathbf{X}_{i}>y\right)=1-\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n}. $
$ f_{\mathbf{Y}_{1}}\left(y\right)=\frac{d}{dy}F_{\mathbf{Y}_{1}}\left(y\right)=n\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n-1}f_{\mathbf{X}}\left(y\right). $
