Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1)
random variables. (
a) Compute the cdf of Y := min(X1, . . . , Xn).
(b) Use (a) to compute the pdf of Y .
(c) Find E(Y ).
Let X1,...,Xn be i.i.d. random variables with mean 0 and
variance 2 > 0. In class we have shown a central limit theorem,
¯ Xn /pn )N(0,1), as n!1 , (1) with the assumption E(X1) = 0. Using
(1), we now prove the theorem for a more general E(X1)=µ6=0 case.
Now suppose X1,...,Xn are i.i.d. random variables with mean µ6= 0
and variance 2. (a) Show that for dummy random variables Yi = Xi µ,
E(Yi) = 0 and V...
Let ? and ? be two independent uniform random
variables such that
?∼????(0,1) and
?∼????(0,1).
A) Using the convolution formula, find the pdf
??(?) of the random variable
?=?+?, and graph it.
B) What is the moment generating function of ??
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1
+ X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y =
(Y1,Y2,Y3)′ using : Multivariate normal distribution
properties.
Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show
that for any α ∈ (0, 1), there is a cn,α, such that
[max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval
of θ.
Let X1,X2,...,Xn be i.i.d. Gamma random variables with
parameters α and λ. The likelihood function is difficult to
differentiate because of the gamma function. Rather than finding the
maximum likelihood estimators, what are the method of moments
estimators of both parameters α and λ?
Let X1,X2,… be a sequence of independent random variables,
uniformly distributed on [0,1]. Define Nn to be the smallest k such
that X1+X2+⋯+Xn exceeds cn=n2+12n−−√, namely,
Nn
=
min{k≥1:X1+X2+⋯+Xk>ck}
Does the limit
limn→∞P(Nn>n)
exist? If yes, enter its numerical value. If not, enter
−999.
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Let X1, ..., Xn be i.i.d random variables with the density
function f(x|θ) = e^(θ−x) , θ ≤ x. a. Find the Method of Moment
estimate of θ b. The MLE of θ (Hint: Think carefully before taking
derivative, do we have to take derivative?)
Let X1, . . . , Xn be a random sample from a uniform
distribution on the interval [a, b]
(i) Find the moments estimators of a and b.
(ii) Find the MLEs of a and b.