Question

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 X^{2}

c) Using the dexp function and the R integrate command calculate
the expected value of X.

d) Using the pexp function find the probability that .2 ≤ X ≤
.4

e) Calculate the probability that X > .2. Use the pexp
function

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.

Answer #1

Question 7) Suppose X is a Normal random variable with with
expected value 31 and standard deviation 3.11. We take a random
sample of size n from the distribution of X. Let X be the sample
mean. Use R to determine the following:
a) Find the probability P(X>32.1):
b) Find the probability P(X >32.1) when n = 4:
c) Find the probability P(X >32.1) when n = 25:
d) What is the probability P(31.8 <X <32.5) when n =
25?...

Question 2) The density of random variable X is f(x) =
15(x2−36)(64−x2) / 3904 for 6 ≤ x ≤ 8 and 0
otherwise. Do computations using the R integrate function.
a) Find the probability that X > 7:
b) Find the probability that 6.5 < X < 7.5:
e) Find the probability that x is within one standard deviation
of its expected value:
f) In the following paste your R script for this problem:

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

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).

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).

Note: The following problem involves the concept that if X~
Exponential(λ) then the expected value (also referred to as average
or mean) of X is 1 / λ. Conversely, λ = 1 / average. Once you know
the value of λ, you also know the PDF and CDF which can be used to
calculate the required probabilities.
The number of days ahead travelers purchase their airline
tickets can be modeled by an exponential distribution with the
average amount of time...

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)...

Question 6) Suppose X is a random variable taking on possible
values 0,2,4 with respective probabilities .5, .3, and .2. Y is a
random variable independent from X taking on possible values 1,3,5
with respective probabilities .2, .2, and .6. Use R to determine
the following.
f) Find the expected value of X*Y. (i.e. X times Y)
g) Find the expected value of 3X - 5Y.
h) Find the variance of 3X - 5Y
i) Find the expected value of...

A random variable XX with distribution
Exponential(λ)Exponential(λ) has the memory-less
property, i.e.,
P(X>r+t|X>r)=P(X>t) for all r≥0 and
t≥0.P(X>r+t|X>r)=P(X>t) for all r≥0 and t≥0.
A postal clerk spends with his or her customer has an
exponential distribution with a mean of 3 min3 min. Suppose a
customer has spent 2.5 min2.5 min with a postal clerk. What is the
probability that he or she will spend at least an additional 2 min2
min with the postal clerk?

If X and Y are independent exponential random
variables, each having parameter λ = 4, find the joint
density function of U = X + Y and
V = e 3X.
The required joint density function is of the form
fU,V (u, v)
=
{
g(u, v)
u > h(v), v >
a
0
otherwise
(a)
Enter the function g(u, v) into the
answer box below.
(b)
Enter the function h(v) into the answer box
below.
(c)
Enter the value...

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