Question

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.

Answer #1

As we are given that X and Y are independent, therefore the value of Y will not depend on any value for X.

Now we are given here that Z = X + Y,

For a given value of X = x, the value of Z is computed as:

Z = Y + x, where x is a constant as it is given.

Also, we know here that:

Therefore the distribution of X would be given as:

Therefore now that we know the conditional X = x, the distribution of Z is uniform, the conditional expected value of Z given X = x, is computed here as:

**Therefore x + 0.5 is the required expected value
here.**

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.

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 .

1. Let P = (X, Y ) be a uniformly distributed random point over
the diamond with vertices (1, 0),(0, 1),(?1, 0),(0, ?1). Show that
X and Y are not independent but E[XY ] = E[X]E[Y ]

Let X, Y ∼ U[0, 1], be independent and let Z = max{X, Y }. (a)
(10 points) Calculate Pr[Z ≤ a]. (b) (10 points) Calculate the
density function of Z. (c) (5 points) Calculate V ar(Z).

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 random variables, uniformly
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Suppose that X is uniformly distributed on the interval [0,5], Y
is uniformly distributed on the interval [0,5], and Z is uniformly
distributed on the interval [0,5] and that they are
independent.
a)find the expected value of the max(X,Y,Z)
b)what is the expected value of the max of n independent random
variables that are uniformly distributed on [0,5]?
c)find pr[min(X,Y,Z)<3]

Suppose that X is uniformly distributed on the interval [0,10],
Y is uniformly distributed on the interval [0,10], and Z is
uniformly distributed on the interval [0,10] and that they are
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a)find the expected value of the min(X,Y,Z)
b)find the standard deviation of the min(X,Y,Z)

Let X be uniformly distributed over (0, 1). Find the density
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a. Y = 6X − 1 b. Z = X2

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 .

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