Question

Given the exponential distribution f(x) = λe^(−λx), where λ > 0 is a parameter. Derive the moment generating function M(t). Further, from this mgf, find expressions for E(X) and V ar(X).

Answer #1

Let X have exponential density f(x) = λe−λx if x >
0, f(x) = 0 otherwise (λ > 0). Compute the moment-generating
function of X.

Poisson Distribution: p(x,
λ) = λx exp(-λ)
/x! , x = 0, 1, 2, …..
Find the moment generating function Mx(t)
Find E(X) using the moment generating function
2. If X1 , X2 ,
X3 are independent and have means 4, 9, and
3, and variencesn3, 7, and 5. Given that Y =
2X1 - 3X2 +
4X3. find the
mean of Y
variance of Y.
3. A safety engineer claims that 2 in 12 automobile accidents
are due to driver fatigue. Using the formula for Binomial
Distribution find the...

Consider a random variable X following the exponential
distribution X ~ f(x), where f(x) = ae^( -ax) for x > 0 and 0
otherwise, a > 0. Derive its moment-generating function MX(t)
and specify its domain (where it is defined or for what t does the
integral exist). Use it to compute the first four non-central
moments of X and then derive the general formula for the nth
non-central moment for any positive integer n. Also, write down the
expression...

Derive the cdf for an exponential distribution with parameter
λ.

Let X be a random variable with probability density function
f(x) = { λe^(−λx) 0 ≤ x < ∞
0 otherwise } for some λ > 0.
a. Compute the cumulative distribution function F(x), where F(x)
= Prob(X < x) viewed as a function of x.
b. The α-percentile of a random variable is the number mα such
that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the
random variable X. The value of mα will...

Problem #3. X is a random variable with an exponential
distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e−λx
for 0 ≤ x where λ = 7.
PLEASE ANSWER these parts if you can.
f) Calculate the probability that X is at least .3 more
than its expected value.Use the pexp function:
g) Copy your R script for the above into the text box
here.

Problem #3. X is a random variable with an exponential
distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e−λx for 0 ≤ x where λ = 7.
a) Using the f(x) above and the R integrate function calculate the
expected value of X.
b) Using the f(x) above and the R integrate function calculate the
expected value of X2
c) Using the dexp function and the R integrate command calculate
the expected value...

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Suppose that X|λ is an exponential random variable with
parameter λ and that λ|p is geometric with parameter p. Further
suppose that p is uniform between zero and one. Determine the pdf
for the random variable X and compute E(X).

If X is an exponential random variable with parameter λ,
calculate the cumulative distribution function and the probability
density function of exp(X).

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