1. The document contains 6 problems related to stochastic processes. The first problem involves finding the distribution, mean, and variance of the sum of random variables where the number of random variables is a Poisson random variable and the random variables themselves are standard normal. 2. The second problem is similar but where the Poisson random variable is truncated. 3. The third problem provides an identity relating the expectation of a weighted compound Poisson random variable to the expectation of another random variable with a shifted distribution.

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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