Homework 1 Due  Feb. 11, 2013 

1. Take the definition of a Poisson process as in Definition 5.1 (p. 313).
Show that 
(a) P( N(t+h) - N(t) = 1 ) = lambda h + o(h)
(b) P( N(t+h) -N(t) >=  2 ) = o(h)


2. The chance for hitting jackpot in a Lotto5 tickets is 1 in 100000.
In a particular day, suppose 150000 tickets were sold. What is the chance there
will be exactly 2 winners of jackpot? 


3. Let N(t) be a Poisson process with rate lambda.  For 0 < s < t find
(a) cov(N(t), N(s))
(b) P(N(s) = 0, N(t) = 3)
(c) E[ N(t)|N(s) = 4]
(d) E[ N(s)|N(t) = 4]



4. If N(t) is a Poisson process with rate lambda, verify that 
N(3t) is also a Poisson process.

If N_1(t) and N_2(t) are two independent Poisson processes, verify that  N_1(t)+N_2(t) is also a Poisson process.