Applied Stochastic process MTH 412 IIT Kanpur

1. Home Work 2
MTH 412
Applied Stochastic Process
1. Let X be a non-negative discrete random variables with possible values 0, 1, 2, . . ..
Show that
E[X] =
∞
n=0
P(X > n) =
∞
k=0
P(X ≥ k).
2. Let X be a non-negative discrete random variables with possible values 0, 1, 2, . . .. Let
the (probability) generating function of X is GX(s) for |s| < 1. Find the P(X = k),
for k = 0, 1, . . ..
3. The following experiment is performed. An observation is made of a Poisson random
variable X with parameter λ. Then a binomial event with probability p of success is
repeated X times and Y success are observed. Find the distribution function of Y .
4. For each fixed λ > 0, let X have a Poisson distribution with parameter λ. Suppose λ
itself is a random variable following a gamma distribution with the following probability
density funciton
f(λ) =
1
Γ(n)
λn−1
e−λ
if λ ≥ 0
0 if λ < 0.
Here n is a fixed positive constant. Find P(X = k), for k = 0, 1, 2, . . . .
5. Suppose Xn’s are independent identically distributed random variables with E|X1| <
∞ and E(X1) = 0. Show that Sn = X1 + . . . + Xn is a martingle.
6. Let X and Y be jointly distributed discrete random variables having possible values
0, 1, 2, . . .. For |s| < 1 and |t| < 1, define the joint generating function
GX,Y (s, t) =
∞
i=0
∞
j=0
si
tj
P(X = i, Y = j).
Prove that X and Y are independent if and only if
GX,Y (s, t) = GX(s)GY (t); for all s, t.
Here GX(s) and GY (t) are the generating functions of X and Y , respectively.
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