Question

Let X i ~ Unif(0, 1) for 1 <= i <= n be IID (independent identically distributed) random variables. Let Y = max(X 1 , …, X n ). What is E(Y)?

Answer #1

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

For X1, ..., Xn iid Unif(0, 1):
a) Show the conditional pdf X(i)|X(j) ∼ X(j)Beta(i, j − i)
b Let n=5, find the joint pdf between X(2) and X(4).

Suppose that X ∼ Unif[0, 3] and Y is independent of X and
exponentially distributed with rate 2.
Find the pdf of
(a) max{X,Y}. (b) min{X,Y}.

Let X and Y be independent and identically
distributed random variables with mean μ and variance
σ2. Find the following:
a) E[(X + 2)2]
b) Var(3X + 4)
c) E[(X - Y)2]
d) Cov{(X + Y), (X - Y)}

Let X and Y be independent, identically distributed standard
uniform random variables. Compute the probability density function
of XY .

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

Let X, Y, and Z be independent and identically distributed
discrete random variables, with each having a probability
distribution that puts a mass of 1/4 on the number 0, a mass of 1/4
at 1, and a mass of 1/2 at 2.
a. Compute the moment generating function for S= X+Y+Z
b. Use the MGF from part a to compute the second moment of S,
E(S^2)
c. Compute the second moment of S in a completely different way,
by expanding...

Let X and Y be independent and identically distributed
with an exponential distribution with parameter 1, Exp(1).
(a) Find the p.d.f. of Z = Y/X.
(b) Find the p.d.f. of Z = X − Y .

a) Let Xi for i = 1,2,...n be random variables with E[Xi] = μi
(not necessarily independent). Show that E[∑ni
=1 Xi] = [∑ni =1 μi]. Show from
Definition
b) Suppose that random variables Yi for i = 1, 2,...,n are
independent and identically distributed withE[Yi] =γ(gamma) and
Var[Yi] = σ2, Use part (a) to show that E[Ybar]
=γ(gamma).
(c) Suppose that random variables Yi for i = 1, 2,...,n are
independent and identically distributed with E[Yi] =γ(gamma) and
Var[Yi]...

For X1, ..., Xn iid Unif(0, 1):
a) ShowX(j) ∼Beta(j,n+1−j)
b)Find the joint pdf between X(1) and X(n)
c) Show the conditional pdf X(1)|X(n) ∼ X(n)Beta(1, n − 1

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