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1. Joint probability density function problem

Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of X, Y, and Z be f(x,y,z) = h(x)h(y)h(z) , x,y,zER Prove that P(X<Y<Z)=1/6 I don't know how to do this at all. This is suppose to be review since this is a...
2. Deduce taylor series

can anyone help me out, i am still stuck in same place
3. Deduce taylor series

I have tried to do this, but I am stuck and cannot get them to equal each other. For f(x)=(1+(-4x))^(-1/2) The taylor series about 0 is: sum (-1/2 choose n) (-4x)^n expanding binomial coefficients: sum -1/2(-1/2-1)(-1/2-2)...(-1/2-n+1) / n! x (-4)^n (x)^n sum -1/2(-3/2)(-5/2)...(1/2-n)/n! x...
4. Deduce taylor series

Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...
5. Variational calculus - dual problem

the primal problem was: min (x^T)Px i found g(r) and the partial derivative of g(r) w.r.t. x to be: x=-1/2(P^-1)(A^T)r i have found the dual problem to be: max -1/4(r^T)A(P^(-1))(A^T)r - (b^T)r subject to r>= 0 I am told to find x* and r* (which i think is just x and r): i have not...

I am looking for radius of convergence of this power series: \sum^{\infty}_{n=1}a_{n}x^{n}, where a_{n} is given below. a_{n} = (n!)^2/(2n)! I am looking for the lim sup of |a_n| and i am having trouble simplifying it. I know the radius of convergence is suppose to be 4, so the lim sup...
7. Showing it is orthogonally diagonalizable

Suppose that the real matrices A and B are orthogonally diagonalizable and AB=BA. Show that AB is orthogonally diagonalizable. I know that orthogonally diagonalizable means that you can find an orthogonal matrix Q and a Diagonal matrix D so Q^TAQ=D, A=QDQ^T. I am aware of the Real Spectral...
8. Finding the function, given the gradient.

ok, so i can get 1/px^p for the x>0 case. but for the x<0 case: i am struggling i have, integ( (-x)^(p-2) x dx) can i write this as: = integ( (-1)^p (x)^(p-2) x dx ) so, = (-1)^p integ (x^(p-2) x dx) which is just = (-1)^p 1/p x^p now how can i put the two together... to make x into |x|?
9. Finding the function, given the gradient.

can you tell me how to integrate this? or at least start, so i can get 1/p|x|^p , i need this small part for a bigger problem and this is making me stuck. i have thought about what you said about the piecewise, but that confuses me even more as i have to deal with not one but 2 functions now
10. Finding the function, given the gradient.

so which way should i be working? gradient f(x) -> f(x)?
11. Finding the function, given the gradient.

ok, should i be working from the gradient f(x) -> f(x) or vice versa. as well , i am getting confused. is this correct: to work from gradient f(x) -> f(x) we integrate. and f(x)-> gradient f(x) we differentiate. working from f(x) -> gradient.. i dont see how i can get gradient f(x). and going...
12. Finding the function, given the gradient.

no the function is this: gradient f(x) = x |x|^(p-2) maybe this is more clear way to write it. and somehow get f(x) = 1/p |x|^p from it.
13. Finding the function, given the gradient.

yes, i have the gradient f(x)= |x|^p-2 x, and i need to find f(x), in class, the definition of gradient is just the derivative w.r.t x of f(x) so i am asking why 1/p |x|^p is the answer because i don't see how you can use this, to find the gradient function |x|^p-2 x. so I thought the function...
14. Finding the function, given the gradient.

the gradient function is |x|^p-2 x and i need to find the function, which apparantly is 1/p |x|^p but i can't figure out how to show this. This is for a bigger problem where the function must be convex. and also p>1 I tried, finding the derivative of 1/p |x|^p , but i don't get the gradient...
15. Gateaux derivative

lim ep->0 <A(u+epv),u+epv> - <Au,u> / ep lim ep->0 <A(u+epv),u>+<A(u+epv),epv>-<Au,u> / ep lim ep->0 <Au,u> + <Aepv,u> + <Au,epv> + <Aepv,epv> - <Au,u> / ep lim ep->0 <Aepv,u> + <Au,epv> + <Aepv,epv> / ep associativity: lim ep->0 ep*<Av,u> + ep<Au,v> + ep*<Av,epv> / ep so ep*(ep) = |ep|^2...
16. Subdifferentiation at a point

the function is not convex. but it is an assignment question so it must be doable some how. so i looked at the wiki you sent me and saw the example. but then i guess it is not the same as my question as that function is convex and mine is not. Example i am referring to: Consider the function...
17. Gateaux derivative

okay, so using your updated associativity rules, my answer is: lim ep->0 <A(u+epv),u+epv> - <Au,u> / ep lim ep->0 <A(u+epv),u>+<A(u+epv),epv>-<Au,u> / ep lim ep->0 <Au,u> + <Aepv,u> + <Au,epv> + <Aepv,epv> - <Au,u> / ep lim ep->0 <Aepv,u> + <Au,epv> + <Aepv,epv> / ep associativity: lim...
18. Gateaux derivative

associativity: <u, av> = <u, av> is there a typo here or are you trying to say that there's nothing you can do to that? ok, i see what i did wrong, here is my revised attempt at the problem. lim ep->0 <A(u+epv),u+epv> - <Au,u> / ep lim ep->0 <A(u+epv),u>+<A(u+epv),epv>-<Au,u> / ep...
19. Gateaux derivative

ok let me try again! no i have not learn this because the teacher expected it to be known as this is a 4th year course and you learn it in the 1st year course (which i am taking at the same time, and this section has not came up yet) my attempt once again, please correct me as I really do not...
20. Gateaux derivative

i'm sorry i don't know anything about inner product, i will be learning it next week for my linear algebra class, but it will be too late for this class as this assignment is due tuesday. if you can inform me about these rules, i will try to apply them and do this problem. Sorry, i missed...
21. Subdifferentiation at a point

Define f:R->R by f(x) = { x^2 sin(1/x) x!=0, 0 x=0 Compute subdifferential f(0) I went through my notes on subdifferentiation and still do not have a clue how to do this process, is there a formula to directly do this? any help would be greatly appreciated.
22. Gateaux derivative

i've never done a question where you plug in an inner product. my attempt would be just this: lim ep->0 <Au+ep,u+ep> - <Au,u> / ep how do i go from there? and i know that the symmetric operator tells u that <Au,u> = <u,Au>
23. Gateaux derivative

suppose A:H->H is a symmetric operator for some Hilbert Space H, define \varphi: H->R by \varphi(u) = <Au,u>. Compute the Gateaux derivative of \varphi I know the definition of the Gateaux derivative, I just don't know how to use the information given to compute it. The definition of the...
24. Variational methods - minimization problem proof

Consider the minimization problem Inf (u E D) F(u) where F(u) = 1/2 integ(0->T) |u (with circle on top)|^2 dt + 1/2 integ(0->T) |u|^2 dt + 1/2 integ(0->T) f(t)u(t) dt, f E L^2 [0,T], and H = {u:[0,T]->R, uEL^2[0,T], u(circle on top) E L^2 [0,T]} is a Hilbert space equipped with the norm...
25. Variational methods - prove f is convex in R->R

Suppose f:R^N -> R is twice differentiable. Prove that f is convex if and only if its Hessian gradiant^2 f(x) is nonnegative. How do I go about proving this? and my professor said I only need to consider when N=1. so R->R. any help would be greatly appreciated. For proving it backwards...
26. Variational methods - properties of convex hull

Show the following properties of convex hull: (a) Co(CoA) = Co(A) (b) Co(AUB) \supseteqCo(A) U Co(B) (c) If A\subseteqB then Co(AUB)=Co(B) (d) If A\subseteqB then Co(A)\subseteqCo(B) The definition of a convex hull is a set of points A is the minimum convex set containing A. (c) is quite...
27. Variational methods conjugate points

(a will be alpha and b will be beta) Let y=y(x,a,b) be a general solution of Euler's equation, depending on two parameters a and b. Prove that if the ratio (subdifferential y/subdifferential a)/(subdifferential y/subdifferential b) is the same at the points, the points are conjugate. I...
28. Variational methods - conjugate of function

Let F:H->R bar be a function and F*:H->R bar its conjugate. Fix aEH and show that the conjugate of the new function G(u)=F(u-a) is G*(u*)=F*(u*)+<a,u> Verify the case where F:R^2->R, F(x)=1/2(x)^2 and a = (2,-1) I don't really know how to show this. please help
29. Conjugate points of extermals of functional

anyone know how to do this?
30. Conjugate points of extermals of functional

Show that the extermals of any functional of the form integ (a->b) F(x,y') dx have no conjugate points. Not sure how to start this question, any help would be appreciated