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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
(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...
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
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
Let T1 be the C-vector space with basis B = (1, cosx, sinx). Define J: T1->T1 by (Jf)(x) = integ(0->pi) f(x-t)dt. Show that J is diagonalizable and find an eigenbasis.
J(1) = integ(0->pi) 1 dt
J(1) = t | (0->pi)
J(1) = pi
J(cos(x)) = integ(0->pi) cos(x-t) dt
J(cos(x)) = - sin (x-t) | (0->pi)...
Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and B are the same.
Some help I have gotten so far but still don't know how to proceed from there:
To prove that the geometric multiplicities of the eigenvalues of A and B are the same, we can...
Let X = C[a,b], J(y) = integ(a to b) sin^3(x) + y(x)^2 dx and D={yEX; integ(a to b) y(x)dx = 1}
(a) what are the D-admissible directions for J?
(b) Find all possible (local) external points for J on D?
so far i have:
(let e be epsilon)
lim e->0 J(y+ev) - J(y) / e
= lim e->0 integ (a to...
Find the adjoint of the operator A:L^2[0,1] -> L^2[0,1] defined by (Af)(x) = integ from (0 to x) f(t)dt
so from my notes it says: the operator A* is called the adjoint of A if:
<Ax, y> = <x, A*y> for all x, vE H
i am not sure how to do this, and need to know how to do it for a test
Let H be a Hilbert space and A: H-> H be a Linear Bounded Operator. Show that A can be written as A=B+C where B and C are Linear Bounded Operators and B is self-adjoint and C is skew.
This is suppose to be an easy question but i'm not sure where to start.
I know that self-adjoint is (B*=B)...
http://img100.imageshack.us/img100/9016/linalggp1.jpg [Broken]
for (a): does that mean i must compute l0(t), l1(t) and l2(t), and i wasn't sure how to do this with the lagrange polynomial formula given, so i found one online and did it, i'm not sure if this is correct, but my l0(t) looks like...
1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)?
Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans.
and for the dimension of sl(n), is...
1. Let X1,X2, ... ,Xn be independent identically distributed random variables with ex-
pected value \mu and variance \sigma^2: Consider the class of linear estimators of the form
\mu\widehat{} = a1X1 + a2X2 + ... + anXn (1)
for the parameter \mu, where a1, a2, ... an are arbitrary constants.
a)...
Give an example of a function f for which \exists s \epsilon R P(s) ^ Q(s) ^ U(s)
P(s) is \forall x \epsilon R f(x) >= s
Q(s) is \forall t \epsilon R ( P(t) => s >= t )
U(s) is \exists y\epsilon R s.t. \forall x\epsilon R (f(x) = s => x = y)
So this was actually a two part question, and...
Compute the line integral \int_{C} F\cdot dr where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1).
i) r(t) = cos t i + sin t j [pi/2 <=t<=...
Suppose that F is an inverse square force field; this is, F(r) = cr/ |r|^{3} for some constant c, where r = xi + yj + zmbfk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the origin.
Not exactly...
Solve the following linear system:
ix + (1+i)y = i
(1-i)x + y - iz = 1
iy + z = 1
I am getting nowhere with this.
is there a trick to do these? I keep getting more and more variations of i. like i^2-1, and (1-i^2)-1
ix + (1+i)y=i
(1-i)x + y-iz=1
y + z = 1
ix + (1+i)y = i
i(1-i)x - (i^2-1)z =...
(a)Find all t \epsilon C such that t^{2} + 3t + (3-i) = 0. Express your solution(s) in teh form x+iy where x,y \epsilon R.
(b) Prove that | 1+iz | = | 1-iz | if and only if z is real.
Okay so I tried to use the quadratic formula to find the roots to find the solutions, but I am stuck...
Find the mass of a right circular cone of base radius r and height h given that the density varies directly with the distance from the vertex
does this mean that density function = K sqrt (x^2 + y^2 + z^2) ?
is it a triple integral problem?
Find the volume of that portion of a unit sphere for which 0<theta<alpha, where theta is one of the spherical coordinates
So i know the equation is z^2 = x^2 + y^2 , but what is the meaning of 0<theta<alpha?
where do i start? I know i must convert to polar coordinates.
z^2 = r^2
Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2
I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.