Question: {Nt}t >= 0, Nt = 0 for t < T1, Nt = 1 for T1 <= t < T1 + T2, ... , where each Ti is exponentially distributed and {Nt}t>= 0 is a Poisson process with rate λ. Prove that Nλn/n converges to 1 in probability as n goes to infinity. Probably have to use weak law of large numbers.
ANSWER
2.1.14. Let X1, X2 be two random variables with joint pmf p(x1, x2) = (1/2)x1+x2 ,
ReplyDeletefor 1 ≤ xi < ∞, i = 1, 2, where x1 and x2 are integers, zero elsewhere. Determine
the joint mgf of X1, X2. Show that M(t1, t2) = M(t1, 0)M(0, t2).
2.2.6. Suppose X1 and X2 have the joint pdf fX1,X2 (x1, x2) = e−(x1+x2)
ReplyDelete, 0 < xi <
∞, i = 1, 2, zero elsewhere.
(a) Use formula (2.2.2) to find the pdf of Y1 = X1 + X2.
(b) Find the mgf of Y1.