Calculate the moment generating functions of the exponential distribution f(x) =1/c (e-x/c), 0≤x≤∞. Hence find its mean and standard deviation.
Answers
Step-by-step explanation:
Demonstrate how the moments of a random variable x may be obtained
from its moment generating function by showing that the rth derivative of
E(ext) with respect to t gives the value of E(xr) at the point where t = 0.
Show that the moment generating function of the Poisson p.d.f. f(x) =
e−µµx/x!; x ∈ {0, 1, 2,...} is given by M(x, t) = exp{−µ} exp{µet}, and
thence find the mean and the variance.
2. Demonstrate how the moments of a random variable may be obtained from
the derivatives in respect of t of the function M(t) = E{exp(xt)}.
If x = 1, 2, 3,... has the geometric distribution f(x) = pqx−1, where q =
1 − p, show that the moment generating function is
M(t) =
pet
1 − qet
.
Find E(x).
Answer: The moment generating function of x is
M(t) = X
∞
x
=1
e
xtpqx−
1
=
p
q X
∞
x
=1
¡
qe
t¢x
= petX
∞
x
=0
¡
qe
t
¢
x
=
pet
1 − qe
t .
To find E(x), we may use the quotient rule to differentiate the expression
M(t) with respect to t. This gives
dM(t)
dt
=
(1 − qet)pet − pet(−qet)
(1 − qet)2
.
Setting t = 0 gives E(x)=1/p.
3. Let xi;i = 1,...,n be a set of independent and identically distributed
random variables. If the moment generating function of xi is M(x
i, t
) =
E{exp(xit)} for all i, find the moment generating function for y = P
x
i
.
Find the moment generating function of a random variable xi
= 0
,
1
whose probability density function if f(xi) = (1
− p)1−x
i p
xi
, and thence
find the moment generating function of y = Pxi. Find
E
(y
) and
V (y).