ECE Ph.D. Qualifying Exam

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.)

Share and discuss your solutions below.

Solution 1

(a) minimum


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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