Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2007
Question
X and Y are iid random variable with
$ P(X=i) = P(Y=i) = \frac {1}{2^i}\ ,i = 1,2,3,... $
a) Find $ P(min(X,Y)=k)\ $.
b) Find $ P(X=Y)\ $.
c) Find $ P(Y>X)\ $.
d) Find $ P(Y=kX)\ $.
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2. (25 Points)
Let $ \left\{ \mathbf{X}_{n}\right\} _{n\geq1} $ be a sequence of binomially distributed random variables, with the $ n $ -th random variable $ \mathbf{X}_{n} $ having pmf $ p_{\mathbf{X}_{n}}\left(k\right)=P\left(\left\{ \mathbf{X}_{n}=k\right\} \right)=\left(\begin{array}{c} n\\ k \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,\cdots,n,\quad p_{n}\in\left(0,1\right). $
Show that, if the $ p_{n} $ have the property that $ np_{n}\rightarrow\lambda $ as $ n\rightarrow\infty $ , where $ \lambda $ is a positive constant, then the sequence $ \left\{ \mathbf{X}_{n}\right\} _{n\geq1} $ converges in distribution to a Poisson random variable $ \mathbf{X} $ with mean $ \lambda $ .
Hint:
You may find the following fact useful:
$ \lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}. $
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