SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? + 2)/(B + 2) for 0 ≤ X < 3, 0 ≤ Y < 3
(Where B=0)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2
SOLUTION REQUIRED WITH COMPLETE STEPS
A continuous random variable X has pdf ?x(?) = (? + 1) ?
2 , 0 ≤ ? ≤ ? + 1, Where B=7.
a) Find the value of a
b) Find cumulative distribution function (CDF) of X i.e.
?x(?).
c) Find the mean of X
d) Find variance of X.
Let X; be n IID U(0, 1) random variables. What are the mean and
variance of the minimum-order and maximum-order statistics?
PLEASE SHOW ALL WORK AND FORMULAS USED
Let X1,…, Xn be a sample of iid random variables with pdf f (x;
?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ.
Determine
a) L(?1, ?2).
b) the MLE of ?⃗ = (?1, ?2).
c) E(? ̂ 2).
Let X1,...,Xn be independent random
variables,and let X=X1+...+Xn be their
sum.
1. Suppose that each Xi is geometric with respective
parameter pi. It is known that the mean of X is equal to
μ, where μ > 0. Show that the variance of X is minimized if the
pi's are all equal to n/μ.
2. Suppose that each Xi is Bernoulli with respective
parameter pi. It is known that the mean of X is equal to
μ, where μ >...
Let X1, X2, . . . be iid random variables following a uniform
distribution on the interval [0, θ]. Show that max(X1, . . . , Xn)
→ θ in probability as n → ∞
Let X1, X2, . . . , Xn be iid Poisson random variables
with unknown mean µ
1. Find the maximum likelihood estimator of µ
2.Determine whether the maximum likelihood estimator is unbiased
for µ
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 X1,…, Xn be a sample of iid N(0, ?) random
variables with Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) the asymptotic variance of the MLE of ?.
d) the MLE of SD(Xi ) = √ ?.