1. Suppose X has a binomial distribution with parameters n and
p. Then its moment-generating function is M(t) = (1 − p + pe^t ) n
.
(a) Use the m.g.f. to show that E(X) = np and Var(X) = np(1 −
p).
(b) Prove that the formula for the m.g.f. given above is
correct. Hint: the binomial theorem says that Xn x=0 n x a^x
b^(n−x) = (a + b)^n .
2. Suppose X has a Poisson distribution with...
Let X1,X2,...,Xn be a random sample from any distribution with
mean μ and moment generating function M(t). Assume that M(t) is
finite for some t > 0.
Let c>μ be any constant. Let Yn = X1+X2+···+Xn. Show that
P(Yn ≥ cn) ≤ exp[−n a(c)] where P(Yn ≥ cn) ≤ exp[−n a(c)]
a(c) = sup[ct − ln M (t)]. t > 0
2. Show that the first derivative of the the moment generating
function of the geometric evaluated at 0 gives you the mean.
3. Let X be distributed as a geometric with a probability of
success of 0.10.
a. Give a truncated histogram (obviously you cannot put the
whole sample space on the x-axis of the histogram) of this random
variable.
b. Give F(x)
c. Find the probability it takes 10 or more trials to get the
first success.
d. Here...
Q4- Describe the moment generating function approach of deriving
a gamma model with mean ?⁄? and variance ? ? 2 ⁄ using n
independent and identically distributed exponential models.?
Describe the moment generating function approach of deriving a
gamma model with mean ?⁄? and variance ? ? 2 ⁄ using n independent
and identically distributed exponential models.
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1/3 (2e ^(3t) +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e^t/( 3 − 2e^t) . Find...
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1 3 (2e 3t +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e t 3 − 2e...
If the moment-generating function of X is M(t) = exp(3 t + 12.5
t2) = e3 t + 12.5 t2.
a. Find the mean and the standard deviation of
X.
Mean =
standard deviation =
b. Find P(4 < X < 16). Round your answer
to 3 decimal places.
c. Find P(4 < X2 < 16). Round
your answer to 3 decimal places.