Given that the joint probability Pr(w,x,y,z) over four variables factorizes as
Pr(w,x,y,z) = Pr(w) Pr(z|y) Pr(y|x,w)Pr(x)
show that x is independent of w by showing that Pr(x,w) = Pr(x)Pr(w).
Attempt: if we simply assume Pr(x,w) = Pr(x)Pr(w), then:
\begin{align}
Pr(w,x,y,z) &= Pr(w)...
1. Use the Singular Value Decomposition (SVD) of G to prove:
rank(XGY^T) = rank (G)
Given that X and Y are two full column-rank matrices, but may not have the same rank.
2. The attempt at a solution
\begin{eqnarray*}
XGY^T & = & X(U\Sigma V^T)Y^T \\
& = & XU \left(...
1. Compute all the values of e^ {\pi i} , indicating clearly whether there is just one or many of them.
Trivially, exp(pi * i) = -1. However, we can also consider e to be the complex number z, and pi * i to be the complex number alpha. Then we get:
e^{\pi i} = z^{\alpha} = e^{\alpha...
Homework Statement
Let f be a one-to-one function from X = {1,2,...n} onto X. Let f k = f(f(f(...f(x))) be the k-fold composition of f with itself. Show that there are distinct positive integers i and j such that f i (x) = f j (x) for all x in X.
Homework Equations
pigeonhole...
Homework Statement
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The potential difference V = 100 V is applied to the capacitor arrangement shown in the figure. Here C1 = 10 microF, C2 = 5 microF, and C3 = 4microF. If capacitor C3 undergoes electrical breakdown (i.e...
Homework Statement
A transverse traveling wave on a cord is represented by D = 0.48 sin (5.6x + 84t) where D and x are in meters and t in seconds. Determine the maximum and minimum speeds of particles of the cord.
The Attempt at a Solution
I'm guessing these speeds are found by the...
Homework Statement
Show that the fractional energy lost per period is
\frac{\Delta E}{E} = \frac{2\pi b}{m\omega_0} = \frac{2\pi}{Q}
where \omega_0 = \srqt{k/m} and Q = m\omega_0 / b
Homework Equations
E = 1/2 k A^2 e^{-(b/m)t} = E_0 e^{-(b/m)t}
The Attempt at a Solution
\Delta E = 1/2 k A^2...
Homework Statement
In some diatomic molecules, the force each atom exerts on the other can be approximated by F = -C/r2 + D/r3, where r is the atomic separation and C and D are positive constants. Let delta r = r - r0 be a small displacement from equilibrium, where delta r << r0. Show that...
A 0.0125 kg bullet strikes a 0.300 kg block attached to a fixed horizontal spring whose spring constant is 2.25 * 10^3 N/m and sets it into vibration with an amplitude of 12.4 cm. What was the speed of the bullet if the two objects move together after impact?
E = .5 k A2 = .5 m v2
Do I use m =...
Homework Statement
A skier of mass m starts from test at the top of a solid sphere of radius r and slides down its frictionless surface. (a) At what angle \theta will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle...
Homework Statement
An airplane pilot fell 370 m after jumping without his parachute opening. He landed in a snowbank, creating a crater 1.1 m deep, but survived with only minor injuries. Assuming the pilot's mass was 80 kg and his terminal velocity was 50 m/s, estimate: (a) the work done by...
The force required to compress an imperfect horizontal spring an amount x is given by F = 150x + 12x3, where x is in meters and F in newtons. If the spring is compressed 2.0m, what speed will it give to a 3.0 kg ball held against it and then released?
I know how to integrate F(x) to get the...
Homework Statement
For two blocks, connected by a cord and sliding down the incline shown in the figure, describe the motion (a) if \mu_1 < \mu_2, and (b) if \mu_1 > \mu_2. (c) Determine a formula for the acceleration of each block and the tension FT in teh cord in terms of m1, m2, and \theta...
Homework Statement
The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is
\psi(r) = (\frac{1}{\pi a_0^3})^{1/2} exp(-\frac{r}{a_0})
a_0 = 0.529 \times 10^{-10} m
(a) What is the probability of finding the electron inside a sphere of volume 1.0 pm3...
"(a) Use the radial wave function for the 3p orbital of a hydrogen atom (see Table 15.2) to calculate the value of r for which a node exists.
(b) Find the values of r for which nodes exist for the 3s wave function of the hydrogen atom."
For part a, I looked at Table 15.2 and found the equation...
Homework Statement
Find the limit.
\lim_{x\rightarrow -\infty} x^2e^x
Homework Equations
L'Hospital's Rule.
The Attempt at a Solution
I rewrite the limit:
\lim_{x\rightarrow -\infty} \frac{e^x}{\frac{1}{x^2}}
applying L'Hospital's Rule:
\lim_{x\rightarrow -\infty}...
Homework Statement
A stunt driver wants to make his car jump over 8 cars parked side by side below a horizontal ramp (Fig. 3-46). (a) With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the cars and the horizontal distance he must clear...
Homework Statement
An unmarked police car traveling a constant 95km/h is passed by a speeder traveling 140 km/h. Precisely 1.00s after the speeder passes, the policeman steps on the accelerator; if the police car's acceleration is 2.00m/s2, how much time passes before the police car overtakes...
I feel that this may be a silly question, but, Does wind affect how far sound can travel? I know that the air temperature affects how far a sound can travel (does wind affect air temperature?), but what about wind? I tried to find an answer via google, but I haven't found anything...
Determine whether f'(0) exists.
f(x) = { x2sin (1/x) if x does not equal 0, 0 if x = 0}
f'(0) = \lim_{x\rightarrow 0} \frac{x^2 \sin \frac{1}{x}}{x - 0}
f'(0) = \lim_{x\rightarrow 0} \frac{\sin \frac{1}{x}}{\frac{1}{x}}
Let u = 1/x. As x approaches 0, u approaches infinity. So I can't use lim...
Find all the points on the graph of the function f(x) = 2 \sin x + \sin^2 x at which the tangent is horizontal.
f'(x) = 2 \cos x + 2 \sin x \cos x = 0
2 \cos x (1 + \sin x) = 0
\cos x = 0 or \sin x = -1
x = \frac{\pi}{2} + n\pi or x = \frac{3\pi}{2} + 2n\pi
I then plug in to find the...
"A metal rod 20 cm long moves in a straight line at a speed of 4.0 m/s with its length parallel to a uniform magnetic field of 1.2 T. Find the resulting potential difference between the ends of the rod."
I don't know how to solve this...
"An alpha particle is accelerated in the +x direction through a voltage of 1000V. The particle then moves in an undeflected path between two oppositely charged parallel plates in a uniform magnetic field of 50 mT in the +y direction. (a) If the plates are parallel to the xy plane, what is the...
"Find the force per unit length on each of two long, straight, parallel wires that are 24 cm apart when one carries a current of 2.0 A and the other a current of 4.0 A in the same direction."
Tell me if I'm doing this right:
I use d = 0.24m, and the equations B = (mag. perm. * I)/(2*pi*d)...
"The magnetic field at the center of a 50-turn coil of radius 15 cm is 0.60mT. find the current in the coil."
N=50, r = .15m, B = 6.0 * 10^-4 T, magnetic permeability = 4 * pi * 10^-7.
B = (mag.per. * N * I)/ L
However, I don't have L, which is the length. I do have r, the radius, but it's...
Could someone please make sure I'm doing this right.
I want to find the derivative of the logarithm to the base a of x, using implicit differentiation.
Let y = \log_{a} x
a^y = x
\frac{d}{dx} (a^y) = 1 (implicit differentiation)
\frac{d}{dx} (e^{\ln a})^y = 1
\frac{d}{dx} (e^{(\ln a)y}) = 1...
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"Find the currents in the circuit branches in [the first figure on the left above]."
V1 = 20V, V2 = 10V, R1 = 5.0 ohms, R2 = 4.0 ohms, R3 = 6.0 ohms...
find the intervals of increase and decrease of f(x) = e3x + e-2x.
f'(x) = 3e3x - 2e-2x
I set f'(x) = 0 to find the critical numbers:
3e3x = 2e-2x
3 ln e3x = 2 ln e-2x
9x = -4x
x = 0, which is obviously wrong, (3e^0 - 2e^0 = 1). I found out that I had to combine the two terms using a...
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"Light bulbs with the wattage ratings shown in the figure above are connected in a circuit. (a) What current does the voltage source deliver to the circuit? (b) Find the power dissipated by each bulb. (Asusume normal operating...