Let ?1, ?2,…. . , ?? (n random variables iid) as a
variable whose pdf is continuous and uniform over the interval [? -
1; ? + 3].
(1) Determine the estimator of the moments method.
(2) Is this estimator unbiased? What is its variance?
(3) Find the maximum likelihood estimator (VME) for this
setting. Is it unique?
Let the random variable Z follow a standard normal distribution,
and let Z1 be a possible value of Z that is representing the 90th
percentile of the standard normal distribution. Find the value of
Z1.
Let N be a binomial random variable with n = 2 trials and
success probability p = 0.5. Let X and Y be uniform random
variables on [0, 1] and that X, Y, N are mutually independent. Find
the probability density function for Z = NXY . Hint: Find P(Z ≤ z)
for z ∈ [0, 1] by conditioning on the value of N ∈ {0, 1, 2}.
Let X1 and X2 be two observations on an
iid random variable drawn from a population with mean μ and
variance σ2 (for example, the random variable could be
income; X1 could be person 1’s income and X2,
person 2’s income). Which of the following is an unbiased estimator
of mean income?
All of these are unbiased estimators of mean income.
23X1+13X2
15X1+45X2
Which of the following is the “best” (that is, least-variance)
estimator of mean income?
15X1+45X2
23X1+13X2
All...
For a random variable Z, that follows a standard normal
distribution, find the values of z required for these probability
values:
P(Z<z)=.5 =
P(Z<z)=.1587 =
P(Z<z)=.8413 =
Please show how to solve without using excel.
Please show all steps.
Thank you!
Let z be a random variable with a standard normal
distribution.
Find “a” such that P(|Z| <A)= 0.95
This is what I have:
P(-A<Z<A) = 0.95
-A = -1.96
How do I use the symmetric property of normal distribution to make
A = 1.96?
My answer at the moment is P(|z|< (-1.96) = 0.95
Let X, Y and Z be independent random variables, each uniformly
distributed on the interval (0,1).
(a) Find the cumulative distribution function of X/Y.
(b) Find the cumulative distribution function of XY.
(c) Find the mean and variance of XY/Z.
Let X denote a random variable that follows a binomial distribution with parameters n=5, p=0.3, and Y denote a random variable that has a Poisson distribution with parameter λ = 6. Additionally, assume that X and Y are independent random variables.
(a) What are the possible values for (X, Y ) pairs.
(b) Derive the joint probability distribution function for X and Y. Make sure to explain your steps.
(c) Using the joint pdf function of X and Y, form...
Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
What are the possible values for (X, Y ) pairs.
Derive the joint probability distribution function for X and Y.
Make sure to explain your steps.
Using the joint pdf function of X and Y, form the summation
/integration...