Question

4. For the following claims, state whether they are TRUE or FALSE. Statements claimed to be TRUE must be accompanied by a proof, and statements claimed to be FALSE must be accompanied by a counterexample.

(a) Let A and B be events in the sample space S. Then P(A ∩ B) ≤ P(A)P(B).

(b) Let E be an event with 0 < P(E) < 1 and define PE(A) = P(A ∩ E) for every event A in the sample space S. Then PE satisfies the axioms of probability

Answer #1

State, with evidence, whether each of the following
claims is true or false:
a. The conditional probability of A, given B, must be
at least as large as the probability of A.
b. An event must be independent of its complement.
c. The probability of A, given B, must be at least
as large as the probability of the intersection of
A and B.
d. The probability of the intersection of two events
cannot exceed the product of their individual...

State, with evidence, whether each of the following
statements is true or false:
a. The probability of the union of two events cannot
be less than the probability of their intersection.
b. The probability of the union of two events cannot
be more than the sum of their individual
probabilities.
c. The probability of the intersection of two events
cannot be greater than either of their individual
probabilities.
d. An event and its complement are mutually exclusive.
e. The individual...

True or false; for each of the statements below, state whether
they are true or false. If false, give an explanation or example
that illustrates why it's false.
(a) The matrix A = [1 0] is not invertible.
[1 -2]
(b) Let B be a matrix. The rowspaces row (B), row (REF(B)) and
row (RREF(B)) are all equivalent.
(c) Let C be a 5 x 7 matrix with nullity 3. The rank of C is
2.
(d) Let D...

State whether the following statements are True or False. If
False, state the correct answer. [4]
The frequency of people with diabetes is a more informative
statistic than the diabetes prevalence rate.
Effective surveillance requires fast action. This is especially
true of chronic non-communicable diseases.

State whether the following statements are true or false.
Statements
True/False
Eurodollars are U.S. dollars deposited in U.S. banks.
▼
False
True
All the bonds denominated in euros are called Eurobonds.
▼
False
True
Eurocurrencies are similar to short-term Eurobonds.
▼
False
True

For each of the following statements, say whether the statement
is true or false.
(a) If S⊆T are sets of vectors, then span(S)⊆span(T).
(b) If S⊆T are sets of vectors, and S is linearly independent,
then so is T.
(c) Every set of vectors is a subset of a basis.
(d) If S is a linearly independent set of vectors, and u is a
vector not in the span of S, then S∪{u} is linearly
independent.
(e) In a finite-dimensional...

Deside whether the statements below are true or false. If
true, explain why true. If false, give a counterexample.
(a) If a square matrix A has a row of zeros, then A is not
invertible.
(b) If a square matrix A has all 1s down the main diagonal,
then A is invertible.
(c) If A is invertible, then A−1 is invertible.
(d) If AT is invertible, then A is invertible.

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if
and only if Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then either X or Y is closed and convex (one or the other).
(iii) Let X be an...

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if
Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then eitherX or Y is closed and convex (one or the other).
(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.
(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then
X∪Y...

State, with explanation, whether the following statements are
True or False:
a) It is possible for ? = ?(?) to have an inflection point at
(?, ?(?)) even if ?′(?) is not defined.
b) If ?''(?) > 0 then ? has a local minimum at ? = ?.
c) For all functions ?, if ?′(?) exists ∀?, then ?′′(?) exists
∀?.

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