Applied Stochastic process MTH 412 IIT Kanpur

1. Home Work 7
MTH 412
Applied Stochastic Process
1. Suppose Xi’s are independent and identically distributed normal random variables
with mean 0 and variance 1, and N is a Poisson random variable with mean λ, find
the distribution function of
Z =
N
i=1
Xi.
Find the mean and variance of Z. Write the distribution of Z as a convex combination
of a discrete and a continuous distribution. Find the probability density function of
the continuous component.
2. Suppose Xi’s are independent and identically distributed normal random variables with
mean 0 and variance 1, and N is a truncated Poisson random variable with parameter
λ, i.e.
P(N = n) = c
λn
n!
; n = 1, 2, 3, . . . .
(i) Find the constant c. Find the distribution function of
Z =
N
i=1
Xi.
Find the mean and variance of Z. Find the probability density function of Z.
3. Let W =
N
i=1
Xi be a compound Poisson random variable and Xi has a PDF f(x), and
N has mean λ. Let X be random variable which is independent of W and it has the
same PDF f(x), then for any function function h(x) where the necessary expectation
exists show that
E(Wh(W)) = λE(Xh(W + X)).
4. Using 3, show that
E(Wn
) = λ
n−1
j=0
n − 1
j
E(Wj
)E(Xn−j
).
5. Show that for a conditional Poisson process with Λ has the following PDF f(λ) = e−λ
for λ ≥ 0 does not have independent increment.
6. Suppose {N(t)} is a conditional Poisson process with Λ follows a gamma distribution
with the shape parameter α and scale parameter λ. Find P(N(t + s) − N(s) = n) for
n = 0, 1, . . . .
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