ECE-QE CS1-2015 - Rhea

Question

Part 1.

If $ X $ and $ Y $ are independent Poisson random variables with respective parameters $ \lambda_1 $ and $ \lambda_2 $, calculate the conditional probability mass function of $ X $ given that $ X+Y=n $.

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Part 2.

Let $ Z(t), t\ge 0 $, be a random process obtained by switching between the values 0 and 1 according to the event times in a counting process $ N(t) $. Let $ P(Z(0)=0)=p $ and

$ P(N(t)=k) = \frac{1}{1+\lambda t}(\frac{\lambda t}{1+\lambda t})^k $

for $ k = 0, 1, ... $. Find the pmf of $ Z(t) $.

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Part 3.

Let $ X $ be an exponential random variable with parameter $ \lambda $, so that $ f_X(x)=\lambda{exp}(-\lambda{x})u(x) $. Find the variance of $ X $. You must show all of your work.

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Part 4.

Consider a sequence of independent random variables $ X_1,X_2,... $, where $ X_n $ has pdf

$ \begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\ &+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align} $.

Does this sequence converge in the mean-square sense? Hint: Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables $ X_1,X_2,... $ converges in mean-square if and only if $ E[|X_n-X_{n+m}|] \to 0 $ as $ n \to \infty $, for every $ m>0 $.

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