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TIGER-Lab 109

Let W(t) be the standard Brownian motion, and 0 < s < t. Find the conditional PDF of W(s = 1/2) given that W(t = 1) = 2. What are the mean and variance? Return the list of [mean, variance].

[1.0, 0.25]

list of float

A scuba diver is wearing a head lamp and looking up at the surface of the water. If the minimum angle to the vertical resulting in total internal reflection is 25∘, what is the index of refraction of the water? $\theta_{air} = 1.00$.

2.37

float

For every positive real number $x$, let $g(x)=\lim _{r \rightarrow 0}((x+1)^{r+1}-x^{r+1})^{1/r}$. What is the limit of $g(x)/x$ as $x$ goes to infinity?

2.7182818

float

Find the orthogonal projection of 9e_1 onto the subspace of R^4 spanned by [2, 2, 1, 0] and [-2, 2, 0, 1].

[8, 0, 2, -2]

list of integer

X rays scattered from rock salt (NaCl) are observed to have an intense maximum at an angle of 20° from the incident direction. Assuming n = 1 (from the intensity), what must be the Wavelength of the incident radiation in nm?

0.098

float

Suppose that there are two firms in the market facing no costs of production and a demand curve given by Q = 150 - P for their identical products. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the nash equilibrium.

0

integer

In how many ways can 10 people be seated at 1 identical round tables? Each table must have at least 1 person seated.

362880

integer

What is the Cramer-Rao lower bound on $E_\theta(\hat{\theta}(X)-\theta)^2$, where $\hat{\theta}(X)$ is an unbaised estimator of $\theta$ for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$.

(b)

option

For an integer a > 0 and an integer b > 0, is there any other number c > 0 such that a^10 + b^10 = c^10?

False

bool

Consider Convolutional Neural Network D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. Consider CNN D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. What is the total number of weights defined for the entire activation output of this first layer? (ie. If you flattened all filters and channels into a single vector)

900

integer

How many ways are there to arrange 9 people in a line such that no one is standing in their correct position?

133496

integer

Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by U(Y) = ln Y. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip’s expected utility?

9.184

float

In how many ways can 6 people be seated at 2 identical round tables? Each table must have at least 1 person seated.

225

integer

For a matrix A, is the function F(A) = det A from the linear space R^{3*3} to R a linear transformation?

False

bool

Are the vectors [1, 2], [2, 3], and [3, 4] linearly independent?

False

bool

Given $V_s = 5V$, $R_1 = 480 \Omega$, $R_2 = 320 \Omega$, and $R_3 = 200 \Omega$, find the power dissipated by the 3 resistors $P_1, P_2, P_3$ in the figure. Represent your answer as a list [$P_1, P_2, P_3$] in the unit of mW.

[51.2, 78.15, 125.0]

list of float

suppose $u=\arctan \frac{y}{x}$, what is numeric of $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}$?

0.0

float

Use the Runge-Kutta method with $h=0.1$ to find approximate values of the solution of $(y-1)^2 * y' = 2x + 3$ with y(1) = 4. What is y(0)?

3.46621207

float

Define f: R o R by f(x) = (x^3) / (1 + x^2). Is f uniformly continuous on R?

True

bool

Evaluate $\int_c z^2 / (z - 5) dz$, where c is the circle that $|z| = 2$.

0

integer

Let a undirected graph G with edges E = {<0,1>,<1,3>,<0,3>,<3,4>,<0,4>,<1,2>,<2,5>,<2,7>,<2,6>,<6,7>,<6,10>,<5,8>,<10,9>,<5,10>,<6,8>,<7,8>,<6,9>,<7,10>,<8,10>,<9,11>,<9,12>,<9,13>,<13,12>,<13,11>,<11,14>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 1 to node 14? Represent the path as a list.

[1, 2, 6, 9, 11, 14]

list of integer

The atomic mass of the 4He atom is 4.002603 u. Find the binding energy of the 4He nucleus in MeV.

28.3

float

Find the ratio of forward-bias to reverse-bias currents when the same voltage 1.5 V is applied in both forward and reverse. Assume room temperature 293 K.

-6e+25

float

In how many ways can a group of 7 people be divided into 2 non-empty subsets?

63

integer

A debt of $25,000 is to be amortized over 7 years at 7% interest. What value of monthly payments will achieve this?

4638.83

float

Assume that half of the mass of a 62-kg person consists of protons. If the half-life of the proton is 10^33 years, calculate the number of proton decays per day from the body.

3.5e-08

float

A ship uses a sonar system to locate underwater objects. Find the wavelength of a 262-Hz wave in water. (Unit: m)

5.65

float

If polygon ABCDE ~ polygon PQRST, AB = BC = 8, AE = CD = 4, ED = 6, QR = QP, and RS = PT = 3, find the perimeter of polygon ABCDE.

30

integer

Are groups Z_4 * Z_2 and D_4 isomorphic?

False

bool

Are the vectors v_1 = [1,2,3], v_2 = [4,5,6], v_3 = [7,8,9] linearly independent?

False

bool

Let M be the inverse of the group element ((3, 5), (4, 6)) in Z_7. What is M[0][1]?

6

integer

In an IPv4 datagram, the value of the total-length field is $(00 \mathrm{~A} 0)_{16}$ and the value of the headerlength (HLEN) is (5) $1_{16}$. How many bytes of payload are being carried by the datagram?

140

integer

How many distinct necklaces with 12 beads can be made with 10 beads of color R and 2 beads of color B, assuming rotations and reflections are considered equivalent?

6

integer

Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3.

7.333

float

The planet Mercury travels around the Sun with a mean orbital radius of 5.8x10^10 m. The mass of the Sun is 1.99x10^30 kg. Use Newton's version of Kepler's third law to determine how long it takes Mercury to orbit the Sun. Give your answer in Earth days.

88.3

float

Consider $x(t)$ to be given as, $$ x(t)=10 \cos (20 \pi-\pi / 4)-5 \cos (50 \pi t) $$ What is minimum sampling rate (/Hz) such that $y(t)=x(t)$ ?

50

integer

Consider an arbitrage-free securities market model, in which the risk-free interest rate is constant. There are two nondividend-paying stocks whose price processes are: $S_1(t)=S_1(0)e^{0.1t+0.2Z(t)}$ $S_2(t)=S_2(0)e^{0.125t+0.3Z(t)}$ where $Z(t)$ is a standard Brownian motion ant $t\ge0$. What is the continuously compounded risk-free interest rate?

0.02

float

Use the Birge-Vieta method to find a real root correct to three decimals of the following equation: x^5 - x + 1 = 0, p=-1.5.

-1

integer

Is the function of f(x) = sin(x) / |x| continuous everywhere?

False

bool

The distortion rate function $D(R)=\min_{p(\hat{x}|x):I(X;\hat{X})\leq R} E(d(X,\hat{X}))$ is convex. True or False?

True

bool

compute the integral \int_{\Gamma} \frac{x*dy-y*dx}{x^2+y^2}, where $\Gamma$ is any piecewise smooth, closed curve that encloses the origin but does not pass through it.

6.2831852

float

suppose the sequence a_n satisfies $lim_{n\rightarrow\infty}a_n\sum_{i=1}^n a_i^2=1$. What is the limit of 3n(a_n)^3?

1.0

float

given a finite group A, and a collection of permutations B. Then (a) there exists B such that A is isomorphic to B; (b) for any B, A is isomorphic to B; (c) A can never be isomorphic to B; (d) none of the above. Which option is correct?

(a)

option

what is the value of $\prod_{n=0}^{\infty}(1+(\frac{1}{2})^{2^n})$?

2.0

float

The diagonals of kite WXYZ intersect at P. If XP = 8, PZ = 8, WP = 6, and PY = 24, find ZY.

25.3

float

Astrophysical theory suggests that a burned-out star whose mass is at least three solar masses will collapse under its own gravity to form a black hole. If it does, the radius of its event horizon is X * 10^3 m, what is X?

8.9

float

Light travel from water n=1.33 to diamond n=2.42. If the angle of incidence was 13 degree, determine the angle of refraction.

7.1

float

An investor has utility function $U(x) = x^{1/4}$ for salary. He has a new job offer which pays $80,000 with a bonus. The bonus will be $0, $10000, $20000, $30000, $40000, $50000, or $60000, each with equal probability. What is the certainty equivalent value of this job offer?

108610

integer

Does \lim_{x \to 0} (cos(mx - 1)/(x^2) = -(m^2)/2 for m = 2?

True

bool

RS is the midsegment of trapezoid MNOP. If MN = 10x+3, RS=9x-1, and PO = 4x+7, what is the length of RS?

26

integer

The bandwidth of an analog signal is 4kHz. An A/D converter is used to convert the signal from analog to digital. What is the minimum sampling rate for eliminating the aliasing problem? (in kHz)

8

integer

Is the cumulative distribution function of the standard gaussian distribution $F(x)=1/\sqrt{2 \pi} \int_{-\infty}^x e^{-t^2/2} dt$ is log-concave? Return 1 for yes and 0 for no.

1.0

float

Compute the mean translational kinetic energy of a single ideal gas molecule in eV.

0.038

float

Let f_1, ..., f_n be polynomials. Do they span the space P of all polynomials?

False

bool

If a cash flow of $100 has a discount rate of 5% and to be received in 5 years, what is the present value of the cash flow?

78.3526

float

What is the value of the integral $\int_0^{\pi/2} 1/(1+(tan(x))^{\sqrt{2}}) dx$?

0.78539815

float

How many labeled graphs with a score of (6, 2, 2, 2, 2, 2, 2) are there?

15

integer

A hydraulic press contains $0.25 m^3$ (250 L) of oil. Find the decrease in the volume of the oil when it is subjected to a pressure increase $\Delta p=1.6 \times 10^7 Pa$ (about 160 atm or 2300 psi). The bulk modulus of the oil is $B=5.0 \times 10^9 Pa$ (about $5.0 \times 10^4 atm$) and its compressibility is $k=1 / B=20 \times 10^{-6} atm^{-1}$. (Unit: 10^{-4} m^3)

-0.8

float

Two sets of points are linearly separable if and only if their convex hulls are disjoint. True or False?

True

bool

A Chord based distributed hash table (DHT) with 25 address space is used in a peer- to-peer file sharing network. There are currently 5 active peers in the network with node ID N3, N8, N15, N19 and N30. Show all the target key (in ascending order, ignore the node's identifier itself) for N3.

[4, 5, 7, 11, 19]

list of integer

What is $\lim _{r \rightarrow \infty} (\int_0^{\pi/2} x^r sin(x) dx)/(r\int_0^{\pi/2} x^r cos(x) dx)$?

0.63662

float

Find the interval in which the smallest positive root of the following equations lies: tan x + tanh x = 0. Determine the roots correct to two decimal places using the bisection method

2.37

float

Is the Fourier transform of the signal $x_1(t)=\left\{\begin{array}{cc}\sin \omega_0 t, & -\frac{2 \pi}{\omega_0} \leq t \leq \frac{2 \pi}{\omega_0} \\ 0, & \text { otherwise }\end{array}\right.$ imaginary?

True

bool

suppose $-\pi<x<\pi$. what is the value of $(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{cos(nx)}{n})/log(2cos(x/2))$? Rounding it to the hundredths place and return the value.

1.0

float

Find the curvature for r(t) = 5cos(t)i + 4sin(t)j + 3tk, t=4\pi/3.

0.16

float

What is 3^(3^(3^(...))) mod 100? There are 2012 3's in the expression.

87

integer

Compute the real integral $I=\int_{-\infty}^{\infty} 1/(x^2 + 1)^2 dx$.

1.57

float

How many different 6-letter arrangements can be made from the letters in the word BANANA?

60

integer

Consider a 26-key typewriter. Suppose that pushing a key results in printing that letter or the next (with equal probability). Thus A results in A or B, ..., Z results in Z or A. What is the capacity of this channel in bits?

3.7

float

Suppose H is a Banach space, and {x_n}\in H, x\in H. Then x_n weakly converges to x is equivalent to: ||x_n|| is bounded; for a dense set M* in H*, it holds \lim_{n\rightarrow\infty} f(x_n)=f(x) for all f\in M*. Is this correct? Answer 1 for yes and 0 for no.

1.0

float

Given 2 colors whose HSI representations are given as follows: which color looks closer to blue? (a) Color 1: $(\pi, 0.3,0.5)$, (b) Color 2: $(0.5 \pi, 0.8,0.3)$

(a)

option

Fig. Q7a shows the amplitude spectrum of a real-value discrete time signal x[n]. Determine the period of signal x[n] (in samples).

8

integer

Consider a 900 Kbytes file stored in a web server. Client A sends a request to the server to retrieve the file from a remote location. There are 3 links (2 intermediate nodes) between server and client and each has a transmission rate of 10Mbps. Given that the segment size is 15 Kbytes, the round trip time (RTT) between the server and client is 30ms, the initial slow-start threshold is 8 and the client's buffer has a storage space of 150 K bytes. Assume that TCP Reno is used, there is no loss during transmission and the headers of protocols are ignored. It is noted that the segments do experience a store-and-forward delay in intermediate routers. Determine how many ms client A takes to receive the whole file from the server after sending a request.

918

integer

Suppose a convex polygon has 26 faces and 39 edges. How many vertices does it have?

15

integer

Given $V_s = 5V$, $R_1 = 480 \Omega$, $R_2 = 320 \Omega$, and $R_3 = 200 \Omega$, find the power dissipated by the 3 resistors $P_1, P_2, P_3$ in the figure. Represent your answer as a list [$P_1, P_2, P_3$] in the unit of mW.

[12, 8, 5]

list of integer

Figure Q8 shows the contour of an object. Represent it with an 8-directional chain code. Represent the answer as a list with each digit as a element.

[6, 7, 0, 6, 6, 4, 3, 4, 3, 1, 1]

list of integer

The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time difference in minutes between the events according to an observer moving at u = 0.8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0.8c.

14

integer

suppose I=[0,1]\times[0,1], where exp is the exponential function. What is the numeric of the double integral of the function f(x,y)=x*y^3 exp^{x^2+y^2} over I?

0.4295

float

Given that $V_A = V_B$, determine the value of $C_2$ (in μF) in the following circuit in the figure.

0.103

float

Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by l = 80w, where l is the number of workers hired and w is their hourly wage. Assume also that Carl's labor demand (marginal revenue product) curve is given by l = 400 - 40MRP_l. How many workers will Carl hire to maximize his profits?

200

integer

A glass contains 0.25 kg of Omni-Cola (mostly water) initially at 25°C. How much ice, initially at -20°C must you add to obtain a final temperature of 0°C with all the ice melted? Neglect the heat capacity of the glass. (Unit: g)

70

integer

The diagonals of rhombus QRST intersect at P. If m∠QTS = 76, find m∠TSP.

52

integer

What is the determinant of the matrix A = [[1, 0, 0, 0, 0, 0], [2, 7, 0, 0, 0, 0], [3, 8, 6, 0, 0, 0], [4, 9, 5, 2, 1, 4], [5, 8, 4, 0, 2, 5], [6, 7, 3, 0, 3, 6]]?

-252

integer

Maximize the entropy $H(X)$ of a non-negative integer-valued random variable $X$, taking values from 0 to infinity, subject to the constraint $E(X)=1$. Use base 2 logarithm to evaluate $H(X)$.

2.0

float

The 4 8x8 images shown below are encoded with JPEG coding. Based on their expected DCT (Discrete Cosine Transform) coefficients, Which image has the most non-zero AC coefficients? (a): Image A, (b): Image B, (c): Image C, (d): Image D.

(b)

option

Your firm is trying to decide whether to buy an e-commerce software company. The company has $100,000 in total capital assets: $60,000 in equity and $40,000 in debt. The cost of the company’s equity is 10%, while the cost of the company's debt is 5%. The corporate tax rate is 21%. What is the WACC of the company?

0.0758

float

is the following function $f(t, y) = \frac{t^3+t^2y+ty+y^3}{t^3 + ty^2}$ scale invariant function

True

bool

In triangle ACD, B is located on the side AC, and E is located on the side AD. If AB = 3, AC = 5, CD = 3.5, ED = 3, and EB ∥ DC, what is the length of AD?

7.5

float

An ultrasonic transducer used for medical diagnosis oscillates at 6.7 Mhz.How long does each oscillation take, and what is the angular frequency? (Unit: 10^7 rad/s)

4.2

float

How many pairs of (a, b) can we substitute for a and b in 30a0b03 so that the resulting integer is divisible by 13?

3

integer

A gun is designed that can launch a projectile of mass 10 kg at a speed of 200 m/s. The gun is placed close to a straight, horizontal railway line and aligned such that the projectile will land further down the line. A small rail car of mass 200 kg and travelling at a speed of 100 m/s passes the gun just as it is fired. Assuming the gun and the car are at the same level, at what angle upwards must the projectile be fired so that it lands in the rail car?

60.0

float

Let x \in R with 0 < x < 1 and n \in N. Is (1 - x)^n >= 1/(1+nx)?

False

bool

Consider a strategy of the form $(\gamma, 0, 0)$ for the investment wheel. Show that the overall factor multiplying your money after $n$ steps is likely to be $(1+2\gamma)^{n/2}(1-\gamma)^{n/2}$. Find the value of $\gamma$ that maximizes this factor.

0.25

float

What is the Fisher information for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$. Which option is correct?

(d)

option

A teacher wants to invest $30,000 into an account that compounds annually. The interest rate at this bank is 1.8%. How much money will be in the account after 6 years?

33389.35

float

Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s

167.0

float

In a set of 20 positive integers, at least how many pairs of numbers have a difference that is a multiple of 10?

10

integer

A symmetric random walk on the three-dimensional cubic lattice Z^3 is transient or persistent? Return 1 for persistent and 0 for transient.

0.0

float

Let C[0,1] be all the continuous function on in the interval [0,1]. For the integral equation $x(t)-\lambda \int_0^1 e^{t-s} x(s) ds=y(t)$, where $y(t)\in C[0,1]$ is a given function. \lambda is a constant and |\lambda|<1. Then there exists a unique solution x(t)\in C[0,1]. This conclusion can be proved by: 1. Implicit function theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the number as the answer.

3.0

float

A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. What fraction of the total energy released goes into the kinetic energy of the proton?

0.000544

float