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diogenes
Questions (10)
Sphere at rest in a uniform stream
---------------------------------------------- Consider a solid sphere of radius a at rest
5 answers
1,155 views
Uniform Flow
------------------------ phi(r, theta) = (A r + B r^(-2) ) Cos(theta) Consider the flow corresponding to n=1, A=U,
1 answer
524 views
Annihilation of a sphere
-------------------------------- A sphere of radius a is surrounded by an infinite mass of liquid
1 answer
634 views
Submarine Explosion
----------------------------- A large mass of incompressible, inviscid fluid contains a spherical bubble
1 answer
586 views
Problem 5. Arrivals during overlapping time intervals
Consider a Poisson process with rate lambda. Let N be the number of
1 answer
2,081 views
What is the probability that an eastbound ship does not pass any westbound ships during its journey through the canal?
0 answers
1,481 views
Suppose that we have three engines which we turn on at time 0.
Each engine will eventually fail, and we model each engine's
1 answer
1,305 views
Exercise: CLT applicability
Consider the class average in an exam in a few different settings. In all cases, assume that we have
2 answers
4,402 views
13. Exercise: Convergence in probability:
a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does
3 answers
5,264 views
Problem 1. Determining the type of a lightbulb.
The lifetime of a type-A bulb is exponentially distributed with parameter 𝜆 .
1 answer
2,414 views
Answers (14)
We consider only the n = 1 mode as described above: the corresponding solution to Laplace's equation is of the form: phi(r, theta) = (A r + B/r^2) cos(theta) We can adjust A and B in order to satisfy the boundary conditions, which are: BC1: v goes to U e_z
We see that here: v = grad(phi) = U e_z and therefore phi = U r cos(theta) is the velocity potential corresponding to the uniform flow of magnitude U in the z-direction.
Solution 1. ------------------- Similarly to the previous example, Bernoulli's equation for unsteady incompressible potential flow under zero body forces takes the form: p(r, t)/ rho + (1/2) (F(t)/r^2)^2 + F'(t)/r + G'(t) == H(t) Letting r go to infinity,
Solution 1. ------------ Here spherical symmetry applies and so: phi(r, t) = (F(t)/r) + G(t) Then we consider our boundary condition. A unit normal to the boundary between the bubble and the fluid is e_r, and so: grad(phi) dot n = (d/d r)(phi) The
3*lambda*exp(- 3*t*lambda)
1. Since students are equally well-prepared and the difficulty level is fixed, the only randomness in a student's score comes from luck or accidental mistakes of that student. It is then plausible to assume that each student's score will be an independent
1. (a) 2. (b) 3. (a) 4. (a)
a) yes b) no c) yes
13. Exercise: Convergence in probability: a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with parameter lambda = 1/n. Does
Sorry, (4) is wrong above: Here are my latest answers: let me know if I am wrong. E[Y|X=x] = -2*x E[Y] = 0 Cov[X,Y]= -2 E[X|Y=y]= (-2*y)/5 Var[X|Y=y]= 1/5
sorry, made a mistake with my Covariance, it is not zero. Any one else get these numbers: (1) -2*x 0 (2) -2 (3) (-2*y)/5 (4) (2*x)+(1/5)
anyone else getting 0 for the Covariance?
In 2019: 1. ln(mu/(3*lambda))/(mu-lambda) 2. (a): (1/4)*e^(-mu*alpha) + (3/4)(1-e^(-lambda*alpha)) 3. 0.3286
In 2019 the answer to part 2 is (a). (1/4)*e^(-mu*alpha) + (3/4)(1-e^(-lambda*alpha))
