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}.$

$f_{\mathbf{Y}_{1}}\left(y\right)=\frac{d}{dy}F_{\mathbf{Y}_{1}}\left(y\right)=n\left(1-F_{\mathbf{X}}\left(y\right)\right)^{n-1}f_{\mathbf{X}}\left(y\right).$

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