Description

Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Discussion 3
1. Triangle Density
Consider random variables X and Y which have a joint PDF uniform on the triangle with vertices at (0,0),(1,0),(0,1).
(a) Find the joint PDF of X and Y .
(b) Find the marginal PDF of Y .
(c) Find the conditional PDF of X given Y .
(d) Find E[X] in terms of E[Y ].
(e) Find E[X].
2. Conditional Distribution of a Poisson Random Variable with Exponentially Distributed Parameter
Let X have a Poisson distribution with parameter λ > 0. Suppose λ itself is random, having an exponential density with parameter θ > 0.
(a) Show that

(b) What is the distribution of X?
(c) Determine the conditional density of λ given X = k, where k ∈N.
3. Poisson Merging
The Poisson distribution is used to model rare events, such as the number of customers who enter a store in the next hour. The theoretical justification for this modeling assumption is that the limit of the binomial distribution, as the number of trials n goes to ∞ and the probability of success per trial p goes to 0, such that np → λ > 0, is the Poisson distribution with mean λ.
Now, suppose we have two independent streams of rare events (for instance, the number of female customers and male customers entering a store), and we do not care to distinguish between the two types of rare events. Can the combined stream of events be modeled as a Poisson distribution?
Mathematically, let X and Y be independent Poisson random variables with means λ and µ respectively. Prove that X + Y ∼ Poisson(λ + µ). (This is known as Poisson merging.) Note that it is not sufficient to use linearity of expectation to say that X +Y has mean λ+µ. You are asked to prove that the distribution of X + Y is Poisson.
Note: This property will be extensively used when we discuss Poisson processes.
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