ECE-QE CS1-2016 - Rhea

Question

Part 1.

A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.

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Part 2.

A point $ \omega $ is picked at random in the triangle shown here (all points are equally likely.) let the random variable $ X(\omega) $ be the perpendicular distance from $ \omega $ to be base as shown in the diagram.
(a) Find the cumulative distribution function (cdf) of $ \mathbf{X} $.
(b) Find the probability distribution function (pdf) of $ \mathbf{X} $.
(c) Find the mean of $ \mathbf{X} $.
(d) What is the probability that $ \mathbf{X}>h/3 $.

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Part 3.

Let $ X $ and $ Y $ be independent, jointly-distributed Poisson random variables with means with mean $ \lambda $ and $ \mu $. Let $ Z $ be a new random variable defined as
$ Z=X+Y $
(a) Find the probability mass function (pmf) of $ \mathbf{Z} $.
(b) Show that the conditional probability mass function (pmf) of $ X $ conditioned on the event $ {Z=n} $ is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters $ "n" $ and $ "p" $) required to specify a binomial distribution $ b(n,p) $).

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Part 4.

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