Wednesday

April 1, 2015

April 1, 2015

Total # Posts: 328

**Probability**

Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, (Y,Z) is uniformly distributed on the set S={(0,0),(0,2),(2,0),(2,2)}. ...
*March 10, 2015*

**Probability**

We are told that events A and B are conditionally independent, given a third event C, and that P(B∣C)>0. For each one of the following statements, decide whether the statement is “Always true", or “Not always true." A and B are conditionally ...
*March 8, 2015*

**Probability**

Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, (Y,Z) is uniformly distributed on the set S={(0,0),(0,2),(2,0),(2,2)}. ...
*March 7, 2015*

**Probability**

Let A,B,C be three events and let X=IA, Y=IB, Z=IC be the associated indicator random variables. We already know that X⋅Y is the indicator random variable of the event A∩B. In the same spirit, give an algebraic expression, involving X, Y, Z for the indicator random...
*March 7, 2015*

**Probability**

A fair, 6-sided die is rolled 6 times independently. Assume that the results of the different rolls are independent. Let (a1,…,a6) denote a typical outcome, where each ai belongs to {1,…,6}. Find the probability that the results of the 6 rolls are all different. (...
*March 7, 2015*

**Probability**

A binary communication system is used to send one of two messages: (i) message A is sent with probability 2/3, and consists of an infinite sequence of zeroes, (ii) message B is sent with probability 1/3, and consists of an infinite sequence of ones. The ith received bit is &#...
*March 7, 2015*

**Probability**

Consider a sequence of independent tosses of a biased coin at times k=0,1,2,…,n. On each toss, the probability of Heads is p, and the probability of Tails is 1−p. A reward of one unit is given at time k, for k∈{1,2,…,n}, if the toss at time k resulted in...
*February 25, 2015*

**Probability**

Each one of n persons, indexed by 1,2,…,n, has a clean hat and throws it into a box. The persons then pick hats from the box, at random. Every assignment of the hats to the persons is equally likely. In an equivalent model, each person picks a hat, one at a time, in the ...
*February 20, 2015*

**Probability**

The newest invention of the 6.041x staff is a three-sided die. On any roll of this die, the result is 1 with probability 1/2, 2 with probability 1/4, and 3 with probability 1/4. Consider a sequence of six independent rolls of this die. 1. Find the probability that exactly two...
*February 20, 2015*

**Probability**

Consider the communication network shown in the figure below and suppose that each link can fail with probability p. Assume that failures of different links are independent. Assume that p=1/3. Find the probability that there exists a path from A to B along which no link has ...
*February 16, 2015*

**Probability**

Before leaving for work, Victor checks the weather report in order to decide whether to carry an umbrella. On any given day, with probability 0.2 the forecast is “rain" and with probability 0.8 the forecast is “no rain". If the forecast is “rain"...
*February 13, 2015*

**Probability**

Oscar has lost his dog in either forest A (with probability 0.4) or in forest B (with probability 0.6). If the dog is in forest A and Oscar spends a day searching for it in forest A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog...
*February 13, 2015*

**Probability**

You roll two five-sided dice. The sides of each die are numbered from 1 to 5. The dice are “fair" (all sides are equally likely), and the two die rolls are independent. Part (a): Event A is “the total is 10" (i.e., the sum of the results of the two die ...
*February 13, 2015*

**Economy**

Consider a competitive firm with a total cost function given by TC(q)=q^2/1000. Suppose that, in order to incentivize higher production, the government decides to refund firms for their cost of producing the FIRST q units produced, up to 1000 of them. What is the LARGEST ...
*February 1, 2015*

**Economy**

Consider a competitive firm that can sell its products at a price of p=10 $/unit. It's variable cost function is given by VC(q)=q^2/1000, and it has SFC=FC=$1000. What is the amount supplied by the firm?
*February 1, 2015*

**Economy**

What is the cost of producing 100 units of output for a firm with a production function given by F(k,l)=10(min(2k,l))^0.5? Assume that r=20 $/unit and w=10 $/unit
*February 1, 2015*

**Neuroscience**

You continue to record the spikes of a ganglion cell. Even in total darkness, there is some nonzero rate of spiking (sometimes called “spontaneous” spiking). When you shine light at a particular point (let’s call it Q) in the retina, the rate of spiking goes ...
*November 26, 2014*

**Neuroscience**

The video compared a retinal neuron to a spy looking through a keyhole at the world. The keyhole is analogous to the 1. lens of the eye 2. pupil of the eye 3. receptive field
*November 26, 2014*

**Neuroscience**

Consider two ganglion cells with cell bodies that are adjacent to each other in the retina. Their receptive fields are 1. overlapping 2. border each other 3. nearby but do not touch
*November 26, 2014*

**Neuroscience**

If you move your eyes to the right, the receptive fields of your neurons will shift in the visual world to the 1.left 2.right 3.up
*November 26, 2014*

**Neuroscience**

The DNA of a neuron is located in its 1. dendrite 2. axon 3. cell body 4. Golgi apparatus
*November 25, 2014*

**Neuroscience**

The longest neurite of a neuron is typically a: 1. cell body 2. dendrite 3. axon 4. none of the above
*November 25, 2014*

**Neuroscience**

A neuron has: 1. a single cell body or soma 2. one or two cell bodies or somata 3. its nucleus in the axon 4. none of the above
*November 25, 2014*

**Mechanical Behavior of Materials**

Consider a sample of well-annealed copper. Find the Burger's vector given that the energy per unit line length is 1.4×10−9J/m, E=177GPa and ν=0.36. Assume α=0.5. b (in nm):
*November 23, 2014*

**Mechanical Behavior of Materials**

What is the exact strain energy per unit length of an edge dislocation for a material that has a Young's modulus E of 50GPa, a Burger's vector b of 0.25nm, a dislocation density ρ of 108cm−2, and a Poisson's ratio ν of 0.3? Uedge (in J/m): What is ...
*November 23, 2014*

**Mechanical Behavior of Materials**

A material has a Young's modulus E of 50GPa, a Burger's vector b of 0.25nm, and a Poisson's ratio ν of 0.3. What is the approximate line tension for a dislocation in this material?
*November 23, 2014*

**Mechanical Behavior of Materials**

help!!!
*November 8, 2014*

**Logic**

Use Propositional Resolution to show that the clauses {p, q}, {¬p, r}, {¬p, ¬r}, {p, ¬q} are not simultaneously satisfiable. Proof takes 8 steps including the premises and the goal.
*October 23, 2014*

**Logic**

Given the premises (p ⇒ q) and (r ⇒ s), use Propositional Resolution to prove the conclusion (p ∨ r ⇒ q ∨ s). Proof takes 8 steps including the premises and the goal.
*October 23, 2014*

**Statics**

Calculate the elastic buckling collapse stress (the compressive strength), σ∗el, for an open-cell polystyrene foam with a relative density of 0.1. (Es for polystyrene = 3.5GPa). Recall that C2=0.05. Give the results in MPa. σ∗el (in MPa): The energy ...
*October 23, 2014*

**Statics**

Compare the Young's modulus E∗ of two open-cell foams. Which Young's modulus is higher? Recall that C1=1. 1. polystyrene: (Es∼3.5GPa), ρ∗/ρs=0.3 2. Al 7075-T651: (Es∼72GPa), ρ∗/ρs=0.1 E∗PS (in GPa): E∗Al (in...
*October 23, 2014*

**Statics**

You have a stainless steel hollow rod (E=193GPa,ν=0.29) that has an inner radius r of 1.2cm, a wall thickness t of 0.7mm, and a length L of 5cm. Calculate the critical loads, in kN, for local buckling and Euler bucking (Pcr,local and Pcr,Euler, respectively). Which would ...
*October 23, 2014*

**Statics**

You have 1kg of aluminum (E=69GPa) to make a cylindrical tube with a high Euler buckling load. The tube must have a length L of 0.5m. You can either make a solid tube, or a thin walled tube with a wall thickness, t=0.5cm. The density of aluminum is 2700kg/m3. The tubes are pin...
*October 23, 2014*

**Statics**

Four identical pin-pin ended solid concrete cylindrical columns that are L=10m tall need to be able to withstand 6.8×108N of force before Euler buckling. What should be the diameter, d, in meters, of each column if the force is uniformly distributed? The Young’s ...
*October 23, 2014*

**Statics**

Calculate the deflection v at the free end, B, for the beams shown below. Both beams have the same rectangular cross section. Give your solution in terms of P,L,E and I.
*October 23, 2014*

**Statics**

A rectangular beam, with cross-sectional width b, height h, Young's modulus Erect, has the same flexural rigidity as a circular beam with the same cross-sectional area. What is the Young's modulus, Ecirc, of the circular beam in terms of Erect,b, and h? What aspect ...
*October 23, 2014*

**Statics**

A cantilever beam, made of a material with a Young's modulus, E with a rectangular cross section with a moment of inertia I, is under a uniformly distributed load, q, across the length L of the cantilever at a point load P at the free end of the cantilever. What is the ...
*October 23, 2014*

**Materials Science**

A hollow cylinder with outer radius r0 and inner radius ri is loaded by a uniformly distributed load q across its length L. The beam is supported by a roller at its left end and a pin L5 from its right end. What is the maximum bending moment in the beam? Express your answer in...
*October 17, 2014*

**Materials Science**

A 4-point bending test is performed on the same beam, with symmetric loads of 5kN separated by 2m. Draw the bending moment and shear force diagrams for the beam. Then answer the following questions about the shear force V and bending moment M. What is the shear force in the ...
*October 17, 2014*

**Materials Science**

A 3-point bending test is performed on a simply supported beam of length L=4m and of rectangular cross-section, with width b and height h. For a central load of 10kN, draw the bending moment and shear force diagrams for the beam. Then answer the following questions about the ...
*October 17, 2014*

**Materials Science**

Calculate the ratio of the moments of inertia I1I2 of two beams of rectangular cross section of the same width, one with a height h and the other, made up of three thinner beams of height h/3. I1/I2 = ?
*October 16, 2014*

**Materials Science**

A 3m long balance beam is simply supported with an overhang at each end of a=0.5m. A gymnast weighing 50kg is standing a distance c to the right of support A and a distance b to the left of support B. How does the maximum bending moment in the beam depend on the distance c? ...
*October 16, 2014*

**Materials Science**

A thin-wall cylindrical tank has radius R, wall thickness t≪R and is subjected to an internal pressure p. The cylindrical wall of the tank is constructed by welding steel plates along 45∘ seams, Consider a small element of the cylindrical wall, and define local ...
*October 10, 2014*

**Materials Science**

At a point in a plate subjected to plane stress loading along its surface, the cartesian stress components acting on the x-face of an element oriented along the (x,y) frame are σx=40MPa and τxy=70MPa, The maximum allowable shear stress, to prevent yielding of the ...
*October 10, 2014*

**Materials Science**

In this problem, you will construct the 2D Mohr's circle for two plane stress states defined in terms of the Cartesian components of the stress tensor in the x,y reference frame: [ σx,σy,τxy]. For each of these stress states, you must construct and use the ...
*October 10, 2014*

**Materials Science**

A compression bar, having a square cross section of side length b, must support a compressive load of magnitude W, uniformly distributed over the bar cross section. The bar is made of two segments connected by a glued joint along plane pq, which is at an angle α to the ...
*October 10, 2014*

**Algorithms**

Arrange the following functions in increasing order of growth rate (with g(n) following f(n) in your list if and only if f(n)=O(g(n))). a)2^2^n b)2^n^2 c)n^2 * log(n) d)n e)n^2^n
*October 9, 2014*

**Networks**

Consider the "Hybrid Model" described in lecture 3.4, with m=10. Let us compare the cases of a=0.8 and a=0.2. At date t=10,000, consider a node born at date i=20 and a node born at date i=9999. Recall that the approximation of the expected degree at time t of a node ...
*October 9, 2014*

**Networks**

Consider the "Preferential Attachment Model" described in lecture 3.3, with m=10. At date t=40, what are: (1) the expected degree for node 15 (born at time i=15), and (2) the expected degree for node 30 (born at time i=30)? [Hint: Use the approximation for the ...
*October 9, 2014*

**Networks**

Consider the model of "Growing Random Networks" described in lecture 3.1 (so the first model where links are added to existing nodes uniformly at random), with m=10. At date t=40, what are: (1) the expected degree for node 15 born at i=15, and (2) the expected degree...
*October 9, 2014*

**Physics**

Which of the following processes is used in order to fabricate monocrystalline silicon? Note that more than one correct answer is possible. The Czochralski process Silicon ribbon process The float-zone process The silicon casting process
*October 8, 2014*

**Physics**

The width of the space charge region at a p-n junction is reduced by applying a voltage bias over the p-n junction. What is the correct statement on both the bias voltage and the p-n junction? The voltage is a forward bias and the diffusion of the majority charge carriers ...
*October 8, 2014*

**Logic**

Consider a propositional language with three propositional constants - mushroom, purple, and poisonous - each indicating the property suggested by its spelling. Using these propositional constants, encode the following English sentences as Propositional Logic sentences. Use ~ ...
*October 6, 2014*

**Big Data**

Suppose ABC is a frequent itemset and BCDE is NOT a frequent itemset. Given this information, we can be sure that certain other itemsets are frequent and sure that certain itemsets are NOT frequent. Other itemsets may be either frequent or not. Which of the following is a ...
*October 6, 2014*

**Big Data**

Imagine there are 100 baskets, numbered 1,2,...,100, and 100 items, similarly numbered. Item i is in basket j if and only if i divides j evenly. For example, basket 24 is the set of items {1,2,3,4,6,8,12,24}. Describe all the association rules that have 100% confidence. Which ...
*October 6, 2014*

**Big Data**

Suppose we want to assign points to whichever of the points (0,0) or (100,40) is nearer. Depending on whether we use the L1 or L2 norm, a point (x,y) could be clustered with a different one of these two points. For this problem, you should work out the conditions under which a...
*October 6, 2014*

**Big Data**

The edit distance is the minimum number of character insertions and character deletions required to turn one string into another. Compute the edit distance between each pair of the strings he, she, his, and hers. Then, identify which of the following is a true statement about ...
*October 6, 2014*

**Big Data**

Suppose our input data to a map-reduce operation consists of integer values (the keys are not important). The map function takes an integer i and produces the list of pairs (p,i) such that p is a prime divisor of i. For example, map(12) = [(2,12), (3,12)]. The reduce function ...
*October 6, 2014*

**Physics**

Let's assume that solar light reaches a silicon solar cell with an angle of incidence of θi=0o. For simplicity, let's consider the refractive index of silicon to be nSi=3.5. The refractive index of air is nair=1. What percentage of light would be lost due to ...
*October 3, 2014*

**Physics**

A solar cell with dimensions 12cm x 12cm is illuminated at standard test conditions. The cell presents the following external parameters: Voc=0.8V,Isc=3A,Vmp=0.75V,Imp=2.5A Calculate the fill factor in percentage [%]: Calculate the efficiency (in %):
*October 3, 2014*

**Physics**

The current density of an ideal p-n junction under illumination can be described by: J(V)=Jph−J0(e^qV/kT−1) where Jph is the photocurrent density, J0 the saturation current density, q the elementary charge, V the voltage, k the Boltzmann's constant, and T the ...
*October 3, 2014*

**Physics**

The width of the space charge region at a p-n junction is reduced by applying a voltage bias over the p-n junction. What is the correct statement on both the bias voltage and the p-n junction? The voltage is a forward bias and the diffusion of the majority charge carriers ...
*October 3, 2014*

**Materials Science**

εx= εy= εz= γyz= γxz= γxy
*October 1, 2014*

**Materials Science**

A cube of the isotropic, linear elastic epoxy resin E=2GPa, ν=0.3, α=50×10−6K−1 at 20∘C is unloaded. It is then heated to 60∘C. What is the corresponding strain matrix? The temperature is held constant at 60∘C. What stress is ...
*October 1, 2014*

**Materials Science**

An isotropic epoxy resin (E=2GPa, ν=0.3) is reinforced by unidirectional glass fibers (Eglass=70GPa), aligned in the 2 direction, such that the fiber composite is transversely isotropic, with the 1-3 plane being the plane of isotropy. The elastic constants of the fiber ...
*October 1, 2014*

**Materials Science**

εx= εy= εz= γyz= γxz= γxy
*October 1, 2014*

**Materials Science**

A linear elastic, isotropic epoxy resin with elastic modulus, E=2GPa, and Poisson’s ratio, ν=0.3, is loaded by the following stress state: σ=⎡⎣10 5 0 5 −20 0 0 0 0⎤⎦MPa Give numerical values for all strain components in the ...
*October 1, 2014*

**Materials Science**

Rubber particles, roughly 1mm in diameter, have been added to an epoxy resin to increase its resistance to crack growth. Estimate the Young's modulus of the composite, given that the Young's moduli of the epoxy and the rubber are, respectively, 2GPa and 10MPa, and that...
*October 1, 2014*

**Materials Science**

Shear modulus G (in GPa):???
*October 1, 2014*

**Materials Science**

Aluminum is a cubic crystal, with S11=1.59×10−11 m2/N S12=−0.58×10−11 m2/N S44=3.52×10−11 m2/N What are the Young's modulus, Poisson's ratio and shear modulus, corresponding to these values? Young's modulus E (in GPa): ...
*October 1, 2014*

**Materials Science**

thx
*October 1, 2014*

**Materials Science**

-8.4 *10^-5 wrong!
*October 1, 2014*

**Materials Science**

A cylindrical pressure vessel, with a diameter of 40cm and a wall thickness of 10mm is made of steel (isotropic, with Young’s modulus = 200GPa; Poisson’s ratio = 0.28) and is pressurized to 20MPa. The longitudinal and hoop stresses in the wall, σl and σh, ...
*October 1, 2014*

**Material Sciences**

The molar crosslink density of a rubber is measured to be 468mol/m3 for a temperature T=300K. What is the Young's modulus of the rubber in MPa? What happens to the modulus of the rubber when the temperature is increased? Young's modulus decreases Young's modulus ...
*September 28, 2014*

**Material Sciences**

Imagine a container is removed from a hot oven and plunged into cold water. Use your answer from Question 3 to justify which material would be more likely to have better thermal shock resistance. Glass: α=9∗10−6C−1 Pyrex: α=4∗10−6C−...
*September 28, 2014*

**Material Sciences**

A steel bar of length L and cross-sectional area A is fitted between rigid supports. Write an expression for the stress in the bar when the temperature is raised by an amount ΔT. Write your answer in terms of α,E and ΔT. Is the stress compressive or tensile? ...
*September 27, 2014*

**Material Sciences**

Given an aluminum shell perfectly bonded to a steel core, what is the thermal stress in the aluminum shell when the system is heated from the unstressed state (20∘C) to 180∘C? Consider only deformations in the axial directions. Give your answer in MPa αAl=23.6...
*September 27, 2014*

**Material Sciences**

The energy-separation curve for two atoms, a distance, r, apart is: U(r)=−A/r^m+B/r^n Derive and expression for the stiffness of the bond at the equilibrium spacing, in terms of A, B, m, n, and r0. S=dF/dr at r=r0:
*September 27, 2014*

**Material Sciences**

The energy-separation curve for two atoms, a distance, r, apart is: U(r)=−A/r^m+B/r^n Derive an expression for the equilibrium spacing, r0, as a function of A, B, m, and n.
*September 27, 2014*

**Material Sciences**

Rubber particles are sometimes dispersed in an epoxy matrix to increase its resistance to the propagartion of cracks. Rubber has a Young's modulus of Erubber=0.001GPa. For a composite with 15 volume percent rubber particles uniformly dispersed in an epoxy matrix, give ...
*September 20, 2014*

**Material Sciences**

A unidriectional, continuously reinforced glass fiber composite has Vf=60 volume percent fibers in a matrix of epoxy. Glass has a Young's Modulus of Eglass=70GPa; epoxy has a Young's Modulus of Eepoxy=2GPa (a) Give numerical values for (i) the Young's Modulus for ...
*September 20, 2014*

**Material Sciences**

The boron fibers are produced by chemical vapor deposition of boron onto fine tungsten wires of dW=10μm. The final wire diameter is dB=75μm. The boron coated wires are then added to an aluminum matrix with VW+B=0.35 fiber volume fraction. The Youngs Moduli of the ...
*September 20, 2014*

**Acoustic**

Consider a room that is 6m long, 5m wide and 3m high and whose walls, floor and ceiling have the same sound absorption coefficient (α=0.01). When all machines are in use, the sound pressure level (SPL) within the room amounts to 90 dB. A carpet is to be placed on the ...
*September 11, 2014*

**Acoustic**

How would you expect the directivity of the piston to vary as the temperature of air decreases? (Hint: Low temperature -> decrease of sound speed) 1.Directivity of the piston is independent of the temperature. 2.Directivity of the piston becomes smooth (low directivity) ...
*September 11, 2014*

**Acoustic**

A breathing sphere of radius 0.1m radiates spherical waves into air at 100Hz with an intensity of 200mW/m2 at the distance 0.5m from the center of the sphere. In this case, 1) Compute the radiated acoustic power. 2) Compute the intensity at the surface of the sphere. 3) ...
*September 11, 2014*

**Music**

List one of the vocal technology topics and describe how it has changed and/or affected popular music. 1. Time Based Effects Processing 2. Autotune or Melodyne 3. Talkbox or Vocoder 4. Vocal Sampling 5. Elastic Audio and Flex
*July 14, 2014*

**Astronomy**

The current peak wavelength of the CMB corresponds to a blackbody of temperature 2.74K. What was the peak wavelength of the CMB when it was emitted? The current redshift of the CMB is z=1100. Give your answer in microns. Which waveband regime is this?
*June 24, 2014*

**Astronomy**

What about a for a supernova in an elliptical galaxy of 10^12 solar masses and radius of 50 kpc? Yes or No
*June 24, 2014*

**Astronomy**

Supernovae can eject material at velocities of 0.001c. Will gas ejected by a supernova which occurs inside a dwarf galaxy with a mass of 10^8 Msolar and a radius of a few kpc be able to escape the galaxy? Yes or No
*June 24, 2014*

**Astronomy**

The star Polaris has a surface temperature of roughly 6500 K and a radius of approximately 50 solar radii. Find the luminosity of Polaris in terms of the Sun's luminosity.
*June 24, 2014*

**Astronomy**

A galaxy is receding away from us with a recession velocity of 100,000 km/s. What is the distance to this galaxy? What is the redshift z of this galaxy?
*June 24, 2014*

**Astronomy**

A quasar is observed at a redshift of z=1, which corresponds to a comoving radial distance of 3 Gpc. Its observed bolometric flux is 10^{-12} erg/cm^2/s. An interacting companion galaxy is seen 4 arcsec apart in projection. What is the bolometric luminosity of the quasar, in ...
*June 19, 2014*

**Maths Probability**

3) 0.0625
*May 20, 2014*

**Probability**

A dedicated professor has been holding infinitely long office hours. Undergraduate students arrive according to a Poisson process at a rate of per hour, while graduate students arrive according to a second, independent Poisson process at a rate of per hour. An arriving ...
*May 13, 2014*

**Probability**

Uncle Henry has been having trouble keeping his weight constant. In fact, during each week, his weight changes from the beginning of the week to the end of the week by a random amount, uniformly distributed between -0.5 and 0.5 pounds. Assuming that his weight change during ...
*May 13, 2014*

**Probability**

Suppose that you move to a new house and you are 10% sure that your new house's phone number is 561290. To verify this, you use the house's phone to dial 561290, obtain a busy signal, and conclude that this is indeed your phone number. (Suppose that you were incorrect...
*May 13, 2014*

**Astronomy**

Imagine that you have found an object out in space. When you use the inverse square law to compute its luminosity, you get a ridiculously large value. You therefore suspect that relativistic beaming is at work - the radiation is coming from a jet traveling at speed v directly...
*May 11, 2014*

**Astronomy**

Imagine that you are an astronomer, and you are planning to do a survey to find new brown-dwarf stars. You need to calculate how sensitive your search needs to be (and hence how big a telescope you will need and how long the survey will take). You want to find ten times more ...
*May 11, 2014*

**Astronomy**

Imagine that you are a particle physicist, and you have come up with a theory that some gamma-ray bursts come from the decay of a new and exotic sub-atomic particle that you think might exist. You calculated that if this particle really does exist and decay as you predict, ...
*May 11, 2014*

**Astronomy**

The luminosity of the Sun is 4*10^33 erg/s, and its radius is 7*10^10 cm. What is the Sun's effective temperature? Please enter your answer in units of Kelvin.
*May 6, 2014*