Homework Statement
In my statistics notes/lectures my professor will oftentimes use an identity that looks like the following:
x_i is a non random variable, the summand is from i=1 to n;
This segment comes from notes on linear regression (y_0 = b_0 + b_1*x_i)
I actually forgot to mention that...
Yes, you are both correct, the correct equation was actually
Equation 1: E(X-bar^2) = (σ^2)/n + μ^2
The mu had to be squared.
In fact, I can see that I have made a huge blunder in how I was looking at this problem. Thank you both for the insight!
Homework Statement
In my notes I keep stumbling upon this equation:
Equation 1: E(X-bar^2) = (σ^2)/n + μ^2
I was wondering why the above equation is true and how it is derived.The Attempt at a Solution
E(X-bar^2)
##Summations are from i/j=1 to n
= E[(Σx_i/n)^2)]
= E[(Σx_i/n)(Σx_j/n)]...
What kinds of things would you guys recommend me to do in order to 'round out my experience?' Are there any sort of classes i should be taking, perhaps a minor that would benefit me? I know that internships should be the first thing I should think of and I will certainly start applying for them...
At the university I was attending I was originally a math major but due to personal circumstance and poor study habits/lack of motivation my GPA was not exactly the best. Due to this and other reasons I was dismissed from univ two years ago but at the moment I am currently in the position of...
Homework Statement
Let ##R## be an ordered relation on a set ##A## that is reflexive and anti-symmetric.
If there is a chain of elements in ##R## that begins and ends with the same element, say the element ##x \in A## is it true that all the elements of ##R## sandwiched in between the ones...
You're right, my case-by-case is actually a contradiction, sorry about that.
But then I'd have to deal with a negative that doesn't really provide much information for an arbitrary partially ordered relation, since the conditions of the relations are undefined. I did attempt it but I ran into a...
Homework Statement
Suppose that R is a partial order on A, B1 ⊆ A, B2 ⊆ A, x1 is the least
upper bound of B1, and x2 is the least upper bound of B2. Prove that if
B1 ⊆ B2 then x1Rx2.
Homework EquationsThe Attempt at a Solution
I split the proof into two different cases:
case 1: x_1 is an...
Kurtz is correct, the given would be of a form
(A or B or C)
Though it doesn't have to have three possibilities, it could have as arbitrarily many, it doesn't really matter for the sake of this discussion, just as long as there are 2 or more.
Homework Statement
If I have a given in a proof in the form:
A or B or C ... etc. etc. and if I choose to approach this given in a case by case basis: (assuming one of the A,B,C... one at a time) and if one or more of the assumptions contradicts some other given in the proof does that mean that...
Homework Statement
Suppose R is a relation on A, and define a relation S on P (A) as follows:
S = {(X, Y ) ∈ P (A) × P (A) | ∀x ∈ X∃y ∈ Y (xRy)}.
For each part, give either a proof or a counterexample to justify your
answer.
(a) If R is reflexive, must S be reflexive?
(b) If R is symmetric...
Homework Statement
Suppose R is a relation from A to B and S and T are relations from
B to C.
Must it be true that (S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R)?
The Attempt at a Solution
I assume that (S \ T ) ◦ R ⊆ (S ◦ R) \ (T ◦ R) is true and attempt to prove it (if I run into a contradiction I...
Thanks for the feedback guys, here's the rest of the proof just for the sake of completeness:
Continuing from the first post:
However, if F IS an empty set then ∅∈F'=∅ and we've found our contradiction; therefore ∃x(x∈A)
Now I prove Uniqueness:
let P(x) = ∃x(x∈A)
We assume P(x) and P(y) and...
Homework Statement
Suppose A is a set, and for every family of sets F, if ∪F = A then
A ∈ F.
Prove that A has exactly one element. (Hint: For both the existence
and uniqueness parts of the proof, try proof by contradiction.)
Homework Equations
The Attempt at a Solution
Let A be...