Question

Let X and Y be independent random variables, uniformly distribued on the interval [0, 2]. Find E[e^(X+Y) ].

Answer #1

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

Let X1,...,X99 be independent random variables, each one
distributed uniformly on [0, 1]. Let Y denote the 50th largest
among the 99 numbers. Find the probability density function of
Y.

Let X and Y be independent random variables, with X following
uniform distribution in the interval (0, 1) and Y has an Exp (1)
distribution.
a) Determine the joint distribution of Z = X + Y and Y.
b) Determine the marginal distribution of Z.
c) Can we say that Z and Y are independent? Good

Let X and Y be independent; each is uniformly distributed on [0,
1]. Let Z = X + Y. Find:
E[Z|X]. Your answer should be a function of x.

(a) Given two independent uniform random variables X, Y in the
interval (−1, 1), find E |X − Y |.
(b) Let X, Y be as in (a). Find the support and density of the
random variable Z = |X − Y |.
(c) From (b), compute the mean of Z and check whether you get
the same answer as in (a)

Suppose that X is a random variable uniformly distributed over
the interval (0, 2), and Y is a random variable uniformly
distributed over the interval (0, 3). Find the probability density
function for X + Y .

Let Y1,Y2.....,Yn be independent ,uniformly distributed random
variables on the interval[0,θ].，Y(n)=max(Y1,Y2,....,Yn)，which is
considered as an estimator of θ. Explain why Y is a good estimator
for θ when sample size is large.

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

X is uniformly distributed on the interval (0, 1), and Y is
uniformly distributed on the interval (0, 2). X and Y are
independent. U = XY and V = X/Y .
Find the joint and marginal densities for U and V .

Let X and Y be random variables, P(X = −1) = P(X = 0) = P(X = 1)
= 1/3 and Y take the value 1 if X = 0 and 0 otherwise. Find the
covariance and check if random variables are independent.
How to check if they are independent since it does not mean that
if the covariance is zero then the variables must be
independent.

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