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...
The probability that A wins B is p. And the probability that B
wins A is q.
If you win 2 more times than opponent, the game is over.
What is the the probability that A two more wins and the game
end?
Simulating a conditional probability using R.
Simulating the conditional probability P(A|B) requires repeated
simulation of
the underlying random experiment, but restricting to trials in
which B occurs.
Suppose 3 dice are tossed and we want to find the probability of
the first die is 4 given that the sum is 10.
Modify the script ConditionalDice.R (you can find the R script file
from Modules on this Canvas site) to find
P(First die is 4 | Sum is 10)
Hint: the...
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 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)
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)