Download Questions on Moment Generating Functions

Questions on Moment Generating Functions
The Basics - practice makes perfect!
1) Find the moment generating functions of the following distributions, then calculate the
mean and variance of these distributions from these MGFs. They should be in order of
difficulty, I hope.
I Binomial random variable with parameter p ∈ [0, 1] (discrete)
II Poisson random variable with λ > 0 (discrete)
III Geometric random variable with parameter p ∈ [0, 1] (discrete)
IV Negative Binomial random variable with parameters p, r ∈ [0, 1] (discrete)
V Exponential random variable with parameter λ > 0 (continuous)
VI Gamma random variable with parameters α, β > 0 (continuous)
VII Gaussian random variable with parameters σ > 0, µ ∈ R (continuous)
VIII Chi-square random variable with parameters p ∈ N (continuous, not in syllabus)
Calculating the MGFs for the Gamma, Gaussian, and Chi-Square is not as simple as just
straightforward integration, there are some tricks involved. You may wish to look at questions
4, 5 and 6 for hints. You may also want to look at the corresponding exam questions 8, 9,
and 10.
2) Find the moment generating function corresponding to
1
a) f (x) = , 0 < x < c.
c
2x
b) f (x) = 2 , 0 < x < c.
c
1 −|x−α|/β
c) f (x) =
e
, −∞ < x < ∞, −∞ < α < ∞, β > 0.
2β
r+x−1
d) P(X = x) =
pr (1 − p)x , x = 0, 1, ..., 0 < p < 1, r > 0 an integer.
x
1
Standard Questions
3) Let the random variable X have a Poisson distribution of parameter λ > 0. Calculate the
moment generating function of X. Suppose that Y is a random variable having the Poisson
distribution of parameter µ and also that X and Y are independent. Use the fact that for
independent X and Y and any functions g, h : R → R,
E[g(X)h(Y )] = E[g(X)]E[h(Y )]
to compute the moment generating function of X + Y . Thus determine the distribution of
X + Y , explaining your reasoning carefully.
4) A lump of radioactive material sends out γ rays which are detected by a Geiger counter.
We wish to compute the probability distribution of the number N (t) of γ photons detected
by time t. The times between detections of γ photons are modelled as independent random
variables T1 , ..., Tn , ..., each following an exponential distribution with parameter λ = 1; that
is having density
(
exp(−t) if t ≥ 0
fT (t) =
0
otherwise
Pn
Let Sn = i=1 Ti be the total time until the detection of the nth γ photon. Show that Sn
follows a distribution with probability density function
(
1
tn−1 e−t if t ∈ (0, ∞)
(n−1)!
fSn (t) =
0
otherwise
by considering moment generating functions. This is known as the Gamma distribution with
scale parameter 1 and shape parameter n.
Next note that
P(Sn > t) = P(N (t) < n).
Use repeated integration by parts to calculate P(Sn > t) and hence show that
P(N (t) = k) = e−t
tk
k!
k ∈ {0, 1, 2, ...}
5) Suppose that X is a random variable with distribution having density
(
1
exp(−x/θ) for x > 0
θ
fX (x ; θ) =
0
otherwise
2
where θ > 0 is a parameter.
i) Compute the moment generating function for this distribution.
ii) Compute the moment generating function of S = X1 + X2 + ... + Xn where X1 , X2 , ..., Xn
are independent and each have the same distribution as X.
iii) Deduce that S has a distribution with density

n−1
 x
exp(−x/θ) for x > 0
(n − 1)!θn
fS (x ; θ) =

0
otherwise
R∞
.
You may use without proof the formula 0 xn−1 e−βx dx = (n−1)!
βn
6) Let Z be a random variable with the standard Gaussian distribution.
a) Write down the density of this distribution and use it to show that the moment generating
function of Z is given by φZ (θ) = exp(θ2 /2).
b) By expressing a random variable X having the Gaussian distribution with mean µ and
variane σ 2 in terms of Z, derive the moment generating function of X.
c) Use moment generating functions to prove that if X and Y are independent Gaussian
2
and σY2 respectively, then X
random variables having means µX and µY and variances σX
2
+ σY2 . State clearly
+ Y has a Gaussian distribution with mean µX + µY and variance σX
any general results about moment generating functions you use.
Past Exam Questions
7) For any k > 0 denote by Xk a random variable having a probability density function fk (x)
that vanishes outside the interval [0, 2] and equals ak xk for 0 ≤ x ≤ 2 with some constant
ak . Calculate the moment generating function G(t) = E[exp(θX1 )) of the random variable
X1 .
8) Normally distributed random variable Y ∼ N (µ, σ 2 ) can be defined by its probability
density function
(
2 )
1 y−µ
1
f (y) = √ exp −
2
σ
σ 2π
Calculate the expectation E[Y ], variance Var[Y ], and the moment generating function φY (θ)
of Y . Of course, they should depend on µ and/or θ.
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9) A random variable Y is called Gamma (θ, n) for θ > 0, n ∈ N if it takes positive values
and has the pdf
y n−1
y
1
exp − , y > 0
fθ,n (y) =
θ(n − 1)! θ
θ
a) Show how to calculate the moment generating function, expectation, and variance of Y .
b) Let X1 , X2 , . . . be independent θ−exponential random variables so that P (Xi > x) =
exp(−x/θ). Find the moment generating function of Xi .
c) Find the moment generating function of Zn = X1 + X2 + . . . + Xn and use it to prove that
Zn is distributed like Gamma (θ, n).
10a) Write down the probability density function of N (µ, σ) of a normal random variable.
Calculate its moment generating function.
b) Let X1 , X2 , . . . be independent N (0, 3) random variables and Zn = X1 + . . . + Xn . Find
the moment generating function of Zn . Is Zn normal? What is the expectation of Z1 00?
11) Write down the pdf of the Gamma (θ, n) random variable X, θ > 0, n ∈ N. Calculate
the moment generating function of this random variable. If X1 , X2 , . . . is a sequence of
i.i.d. random variables distributed each like Gamma (θ, n), determine the sequences bk and
ak , k ∈ N, such that the sequence of the normalized sums
X 1 + . . . + X k − ak
bk
is asymptotically standard normal.
(You need to use the Central Limit Theorem as well.)
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