In: Statistics and Probability

Using the axioms of probability, prove:

a. P(A U B) = P(A) + P(B) − P(A ∩ B).

b. P(A) = ∑ P(A | Bi) P(Bi) for any partition B1, B2, …, Bn.

Prove using only the axioms of probability that if A and B are
events, then P(A ∪ B) ≤ P(A) + P(B)

Assume B is a Boolean Algebra. Prove the following statement
using only the axioms for a Boolean Algebra properties of a Boolean
Algebra.
Uniqueness of 0: There is only one element of B that is an
identity for +
please include all the steps.

Using field axioms and order axioms prove the following theorems
(explain every step by referencing basic axioms)
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions...

Using the standard normal distribution, find each
probability.
a) P(0 < z < 2.23)
b) P (-1.75 < z < 0)
c) P (-1.48 < z < 1.68)
d) P (1.22 < z < 1.77)
e) P (-2.31 < z < 0.32)

Prove the following theorem. Using the ruler function axiom. List
all axioms and definitions used.
Let P and Q be two points, then the line segment AB=BA (AB and
BA have lines over them to show line segments)

Using field axioms, prove the following theorems:
(i) If x and y are non-zero real numbers, then xy does not equal
0
(ii) Let x and y be real numbers. Prove the following
statements
1. (-1)x = -x
2. (-x)y = -(xy)=x(-y)
3. (-x)(-y) = xy
(iii) Let a and b be real numbers, and x and y be non-zero real
numbers. Then a/x + b/y = (ay +bx)/(xy)

Please justify and prove each statement (Use explicitly the four
axioms)
a) Prove that a finite positive linear combination of metrics is
a metric (Use explicitly the four axioms). If it is infinite, will
it be metric?
b) Is the difference between two metrics a metric?
(d1 - d2)

prove or disprove using logical equivalences
(a) p ∧ (q → r) ⇐⇒ (p → q) → r
(b) x ∧ (¬y ↔ z) ⇐⇒ ((x → y) ∨ ¬z) → (x ∧ ¬(y → z))
(c) (x ∨ y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ⇐⇒ ¬y → (x ↔ z)

Prove or disprove using a Truth Table( De Morgan's Law) ¬(p∧q) ≡ ¬p∨¬q Show the Truth Table for (p∨r) (r→¬q)

Prove that if U, V and W are vector spaces such that U and V are
isomorphic and V and W are isomorphic, then U and W are
isomorphic.

ADVERTISEMENT

ADVERTISEMENT

Latest Questions

- Matlab Code Write a procedure to calculate the log discriminant function for a given multi-variate Gaussian...
- The table shows the prices (in dollars) for a sample of automobile batteries. The prices are...
- a. The following balance sheet and information are given for Mancini International Corporation (MIC): Current Assets...
- Farmers and foresters often inoculate seeds with fungal spores to promote plant growth and development. Based...
- Calculate Payback period, Net Present Value and Benefit Cost Ratio Period 0 1 2 3 4...
- For the following projects, compute the net present value, internal rate of return, and the profitability...
- The acrosstown company has an equity beta of 0.5 and 50% debt in its capital structure....

ADVERTISEMENT