I have some homework i'm a lil stuck on right now for my probability class.

it deals with joint probability mass functions

given that the joint density function is fx,y(x,y)= 1/x, for 0<y<x, 0<x<1
find fy|x=0.8(y) and E(Y|X = 0.8)

as well as this question that has me lost as well
the joint p.m.f of X and Y is given as
px,y(1,1) = 1/8, px,y(1,2) = 1/4, px,y(2,1) = 1/8, px,y(2,2) = 1/2
Determine E(X|Y = 1) and Var(X|Y = 1)
Compute P(X|Y <= 1)

any help would be greatly appreciated

This is what I thought. But I'm starting Evans' PDE, and in Appendix A on notation, he has: u_x1x2x3 = d^3 u/dx1 dx2 dx3. I'm sure the meaning will be clear in context. But it seems counter-intuitive.

Perhaps he will only use this notation with C^∞ functions? (In that case the order wouldn't matter, correct?)

Okay I was just going by textbooks I've seen. Maybe it's not totally standard. You do know Clairaut's Theorem, right?

Yes, I've seen the proof for the mixed 2nd-order partials of a C² function.

And I hate to ask this explicitly, but I'm just so clueless: given a C^k function f, and a multiindex α where |α| = k, D^αf is indifferent to the order of all k partials, right?

Hey guys,
I'm back at it and think I'm making progress with induction but am stuck on this one:

Prove: (m*n)*k = m*(n*k)

Base Case: (m*n)*1 = m*(n*1), m*n = m*n, we're given that 1*m = m and m*(n+k) = m*n + m*k

Induction Hypothesis: (m*n)*k = m*(n*k)
Prove: (m*n)*(k+1) = m*[n*(k+1)]

(m*n)*(k+1) = [(m*n)*k] + m*n (distributive law)
= [(m*(n*k)] + m*n (induction hypothesis)
= m*[(n*k)+n] (distributive law)
= m*[n*(k+1)] (distributive law)

Is this right and is my justification reasonable? I think we're just using the distributive law in reverse (undistributing things)...

Thanks guys

Thumbs up.

I don't believe that this method of proof shows anything. What is the original question? I can't imagine ever proving associativity by induction.

Hey guys,

struggling with intergration and geometric series calculations.

firstly my question is integrate(sorry cant find symbol) xcos2x dx= ....

I was thinking the answer should all be over 2 or have x squared in it but not really sure to a solution, maybe x^2cos/2

also with geometric series I dont understand how to find infinity when we are calculating the sum of a series.

Thanks in advance

You have to use integration by parts, which is a rule that says that for function and we have that where denotes the primitive of

When using this rule it's worth it to think about what function to choose as and what function as .

In your case, note that if one would choose and the integral isn't exactly easier to calculate than the one you started with (which is something you want when using integrating by parts).

However, if we pick and we see that in the integral on the right hand side that and so we find

How else would you prove that integer multiplication is associative?

Doing a quick review of my old Halmos "Naive Set Theory", I see the principle of induction is pretty much The primitive defining property of the natural numbers considered as a set - prior even to the defintion of addition.

The development builds on the Axiom of Infinity whereby we assume the existence of a "Successor Set" containing 0 ( 0 identified with the empty set). A Successor Set, S, is characterized by the property: x in S implies (x union {x}) is in S. (Once addition is defined for a natural number x, the successor is identified with x+1).

The Set of Natural Numbers, N, is then defined to be the minimal such Successor Set containing 0. So this is before addition is defined. And the minimal propery amounts to the principle of induction. ie. If S is another successor set containing 0 which is contained in N then S must be N. Otherwise it would be smaller than N contradicting the minimal property for N.

Addition on N is then defined according to this induction principle with all the basic arithmetic properties flowing from it as well. Once the successor set of x is identified with x+1 you can see that S being a Successor Set containing 0 is equivalent to satisfying the conditions of an induction proof. 0 is in S, and x+1 is in S whenever x is in S. Therefore S must be N. That's the minimal Succssor Set defining property of the set of Natural Numbers.


PairTheBoard

Thanks for the confirmation, Wyman.

If we want to prove that the absolute value of sin(nx) <_ n*absolute value of sin(x), my book says to be careful not to confuse the general statement to be proved and the "fixed case of the inductive step."

prove: abs(sin(nx))<_n*abs(sin(x)), for all x and n

whereas the fixed case of the inductive step: abs(sin(kx))<_k*abs(sin(x)) for k fixed, all x

I don't really understand the clarification here. How do we know to induct on k instead of to induct on x...or how do we know which variable is being held fixed?

In earlier cases this wasn't an issue because we only had one variable to work with. Unfortunately, my book isn't very clear on this point.

Thanks again for all the help guys

tyty

Hey Mariogs, your proof does assume the Axiom of Infinity, right?

@lastcard: not sure what you mean by that. The book defines addition and multiplication and goes over the peano axioms...

Why does Munkres define a k-manifold in ℝⁿ as a subspace of ℝⁿ? Doesn't a subspace M have the property that if x ∈ M, then cx ∈M for any scalar c?

None of his example manifolds seem to have this property...

The role the Peano axioms play is in the definition of "natural number" in the theorem that natural number multiplication is associative. As PTB said, this definition assumes the Axiom of Infinity. Informally, that axiom says that the set {0, 1, 2, ..} exists, and then a natural number is defined as any element of this set.

If you rejected the Axiom of Infinity you could still, arguably, have a working definition of natural number, in the form of a rule for generating the sequence 0, 1, 2, ... Such as how children are taught and computers are programmed. Then you could use induction to prove that multiplication as defined in this context is associative.

My point is that inductive reasoning is in either case essentially the same, and so seems quite independent of the Axiom of Infinity.

Theorem: For all real x, and for all natural n, abs(sin(nx)) <= n*abs(sin(x))
Proof: By induction on n.
Base case (n=0): For this case, we must show that for all real x, abs(sin(0*x)) <= 0 * abs(sin(x)). This is patently obvious, since both sides are 0.
Inductive Hypothesis: Assume that for some natural number n* and all real x, abs(sin(nx)) <= n*abs(sin(x)) [*note that we may call this number k instead to avoid ambiguity].
We would like to show that for all real x, assuming the inductive hypothesis, abs(sin((n+1)x)) <= (n+1)*abs(sin(x)). The theorem would then follow from the principle of mathematical induction.

We know that sin(a+b) = sin(a)cos(b)+cos(a)sin(b).

abs(sin((n+1)x))
= abs( sin(nx)cos(x) + cos(nx)sin(x) ) [applying the trig identity above]
<= abs( sin(nx)cos(x) ) + abs( cos(nx)sin(x) ) [triangle inequality]
= abs(sin(nx))abs(cos(x)) + abs(cos(nx))abs(sin(x))
<= abs(sin(nx)) + abs(sin(x)) [since abs(cos()) <= 1 for all ]
<= n*abs(sin(x))+ abs(sin(x)) [inductive hypothesis]
= (n+1)*abs(sin(x)),
proving the theorem.

Alright, I've got a question about degrees of freedom for a t-distribution.

Say I've got a population with 29 clusters, and 4 of them are taken by SRS. Then, within each cluster, another SRS is taken to select elements within the cluster. A question is asked and I have the data of how many in each SRS answered yes to the question. I'm trying to find the proportion who would answer yes in the entire population.

I've found an estimate for the proportion and the variance of that estimate. But for, say, a 95% confidence interval, I'm not entirely sure what t* value I should use to calculate the bound. I've got a t-table with values for different degrees of freedom, but I'm a little confused on which to actually go with. Halp?

Hey if Thorium-227 has a half-life of 18.4 days. How much time will a 50-mg sample take to decompose to 10mg. I got to 5^-1=2^-t/18.4 and i am not sure how to get the bases to equal each other to solve. Any help would be great, thanks.

Forgive me if something like this has been posted, but I did a search and couldn't find it...

Anyways...

A firms' Avg Cost Function is f(q) = 200 +48q^2

I need to find the Total Cost function and Marginal Cost function and explain in a couple sentences how I got each of them and what they mean.

I'm guessing TC at 100 units produced is...

200 + 48(100)^2

which is 482000 ?

And MC is 96 (The derivative of the ATC function)

My question is...what do these two things mean?

Any serious LaTeX gurus in here? I'm having trouble with some new package installation. I'm trying to install some packages that are not on ctan. If you'd be willing to help me thru it via a 5 min AIM convo, please PM -- I'd really appreciate it. [I use TeXnicCenter and MikTeX on windows, but if you're more familiar, my computer dual-boots ubuntu, and I'm installing texlive-full right now.]

TIA.

take your equation and simplify it, remember if a = b^x

then x = log_b(a)

so in your case a = 0.2, b = 2 and x = -t/18.4

-t/18.4 = log₂(.2) since your caluculator may not do logs with a base of 2 here is an easy way to calculate it...

log₂(n) where n is some number is equal to ln(n)/ln(2)

log₂(.2) = ln(.2)/ln(2) = -2.322

you're left with -t/18.4 = -2.322, solve for t and get 42.7 days

As I read the Axiom of Infinity, if I were to state it informally I believe it says, There exists a set containing 0 with the property that x+1 is in the set whenever x is in the set. That looks to me almost an Axiom of Induction. I think it then does become essentially an assumption of the Induction Principle when N is taken to be the unique Minimal set of this kind. In other words, assuming the axiom of infinity amounts to assuming the principle of induction which also amounts to assuming the existence of the natural numbers.


PairTheBoard

To simplify my last response to this. Induction is the defining property of the Natural Numbers. So it should not be suprising that other properties of the Natural Numbers - like associativity - should be proved by way of the defining property of the Natural Numbers, namely by Induction.


PairTheBoard