Question

If X and Y are correlated and Y and Z are correlated, then X and Z are correlated.

prove or disprove?

Answer #1

**Disprove**

**Correlation is not Transitive:**

**If** X and Y are correlated and Y and Z are
correlated, then it will not always be the case that X and Z are
correlated. We have to check the corrleation between variables ,
their strenght and direction.

The single number (the correlation coefficient) does not
encapsulate enough information to guarantee transitivity of the
property of being positively correlated. However, if we know the
correlation coefficient rXY and rYZ then we can find upper and
lower bounds for rXZ .

**THANKS**

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Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Prove or disprove following by giving examples:
(a) If X ⊂ Y and X ⊂ Z, then X ⊂ Y ∩ Z
(b) If X ⊆ Y and Y ⊆ Z, then X ⊆ Z
(c) If X ∈ Y and Y ∈ Z, then X ∈ Z

Let X = { x, y, z }. Let the list of open sets of X be Z1. Z1 =
{ {}, {x}, X }. Let Y = { a, b, c }. Let the list of open sets of Y
be Z2. Z2 = { {}, {a, b}, Y }.
Let f : X --> Y be defined as follows: f (x) = a, f (y) = b,
f(z) = c
Is f continuous? Prove or disprove using the...

X, Y, Z are zero mean correlated random variables with common
correlation coefficient equal to - 1/2 and all variances equal to
one.
a. Find the best linear estimate of Z in terms of X and Y ?
b. Find the best linear estimator for X in terms of Y and Z?
c. What are the minimum mean square estimation errors in the
above cases?

Prove or disprove the following statements.
a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k,
then gcd(a, b) = k
b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.

2. Define a function f : Z → Z × Z by f(x) = (x 2 , −x).
(a) Find f(1), f(−7), and f(0).
(b) Is f injective (one-to-one)? If so, prove it; if not,
disprove with a counterexample.
(c) Is f surjective (onto)? If so, prove it; if not, disprove
with a counterexample.

Prove or disprove that there do not exist z, y,
and z are positive integers such that X7 - Y5
= Z4

3. Consider the statement ∀x ∈ Z ∃y ∈ Z : (x 6= y) ∧ (x|y). (a)
Negate this statement. The statement you obtain may not contain the
symbol ¬. (b) Write the original statement in English without using
any symbols or variables! (c) Is the original statement true? Prove
your answer.

Simplify this expression
Q = (x+y’+z)(x+y’+z’)(x’+y+z)(x’+y’+z)
D = (x’+y’+z’)(x’+y+z’)(x’+y+z)

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