Question

Let Ω = {0, 1}^3 , that is, all possible (ordered) triples of zeros and ones.

Suppose that all outcomes have equal probability.

We define three random variables X1, X2, and X3 on this space representing the first, second, and third digit, respectively.

We also define X = X1 + X2 + X3.

(i) Compute the values (across Ω) of each of the following random variables: E(X|X1), E(E(X|X1)|X2), E(X2|X).

(ii) What is the probability mass function of E(X2|X).

Answer #1

Let X1 and X2 have the joint pdf
f(x1,x2) = 8x1x2 0<x1 <x2 <1
0. elsewhere
What are the marginal pdfs of x1 and x2?
Find the expected values of x1 and x2.
3. What is the expected value of X1X2? (Hint: Define
g(X1, X2) = X1X2 and extend the definition of expectation of
function of a random variable to two variables as follows: E[g(X1,
X2)] = ? ? g(x1, x2)f(x1, x2)dx1dx2.
4. Suppose that Y = X1/X2. What...

1. An electronic system has two different types of components in
joint operation. Let X1 and X2 denote the
Random Length of life in hundreds of hours of the components of
Type I and Type II (Type 1 and Type 2), respectively. Suppose that
the joint probability density function (pdf) is given by
f(x1, x2) = { (1/8)y1
e^-(x1 + x2)/2, x1 > 0,
x2 > 0
0 Otherwise.
a.) Show that X1 and X2 are
independent.
b.) Find E(Y1+Y2)...

1)Let X1, ..., Xn be independent standard normal random
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7...

There are two traffic lights on a commuter's route to and from
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X2 be the number of lights at which he must
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independent, each with the pmf given in the accompanying table (so
X1, X2 is a random sample
of size n = 2).
x1
0
1
2
μ...

2. Probability (30%). Figure out the probability in the
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Suppose you will draw 3 marbles without replacement from each of
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10 red marbles and 20 black marbles; bag 3 has 500 red marbles and
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Let n be a positive integer and p and
r two real numbers in the interval (0,1). Two random
variables X and Y are defined on a the same
sample space. All we know about them is that
X∼Geom(p) and
Y∼Bin(n,r). (In particular, we do not
know whether X and Y are independent.) For each
expectation below, decide whether it can be calculated with this
information, and if it can, give its value (in terms of p,
n, and r)....

Let ?(?)=2?3−9x^2+3?+4.
List all possible rational roots of ?g, according to the
Rational Zeros Theorem.
You must list positive and negative roots separately (there is no
±± in WeBWorK).
Separate your list with commas.
Factor ? completely:
?(?)=____
The x-intercepts of the graph of ?=?(?) are:
Note: Use commas to separate your answers.
?=___
The y-intercept of the graph of ?=?(?) is:
?=____
Select the correct graph of ?=?(?): ? Graph A Graph
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Let U1 and U2 be independent Uniform(0, 1) random variables and
let Y = U1U2.
(a) Write down the joint pdf of U1 and U2.
(b) Find the cdf of Y by obtaining an expression for FY (y) =
P(Y ≤ y) = P(U1U2 ≤ y) for all y.
(c) Find the pdf of Y by taking the derivative of FY (y) with
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(d) Let X = U2 and find the joint pdf of the rv pair...

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All numbers have an equal probability of being selected. Find the
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(a) P(x≤a)=0.8
a=
(b) P(x < a) = 0.25
a=
(c) P(x≥a)=0.17
a=
(d) P(x>a)=0.73
a=
(e) P(0.15≤x≤a)=
a=

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