diogenes

Problem 1. Determining the type of a lightbulb. The lifetime of a type-A bulb is exponentially distributed with parameter 𝜆 . The lifetime of a type-B bulb is exponentially distributed with parameter 𝜇 , where 𝜇>𝜆>0 . You have a box full of lightbulbs of...

6 years ago

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. Doe...

6 years ago

Exercise: CLT applicability Consider the class average in an exam in a few different settings. In all cases, assume that we have a large class consisting of equally well prepared students. Think about the assumptions behind the central limit theorem, and...

6 years ago

Suppose that we have three engines which we turn on at time 0. Each engine will eventually fail, and we model each engine's lifetime as exponentially distributed with parameter lambda. One of the engines will fail first, followed by the second, and follow...

6 years ago

Problem 5. Arrivals during overlapping time intervals Consider a Poisson process with rate lambda. Let N be the number of arrivals in the interval from 0 to t. Let M be the number of arrivals in the interval from 0 to (t+s). t is greater than 0, s is grea...

6 years ago

Submarine Explosion ----------------------------- A large mass of incompressible, inviscid fluid contains a spherical bubble obeying Boyle's Law: p V = constant At great distances from the bubble, the pressure is zero. Neglecting body forces, show that th...

6 years ago

Annihilation of a sphere -------------------------------- A sphere of radius a is surrounded by an infinite mass of liquid modeled as an ideal fluid of mass density rho. The pressure at infinity is Pi. The sphere is suddenly annihilated at t==0. Show that...

6 years ago

Uniform Flow ------------------------ phi(r, theta) = (A r + B r^(-2) ) Cos(theta) Consider the flow corresponding to n=1, A=U, B=0: phi(r, theta) = U r Cos[theta] = U z

6 years ago

Sphere at rest in a uniform stream ---------------------------------------------- Consider a solid sphere of radius a at rest with its center being the origin of the system (r, theta, curly-phi). The sphere is immersed in an infinite stream of an ideal fl...

6 years ago

In 2019 the answer to part 2 is (a). (1/4)*e^(-mu*alpha) + (3/4)(1-e^(-lambda*alpha))

6 years ago

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

6 years ago

anyone else getting 0 for the Covariance?

6 years ago

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)

6 years ago

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

6 years ago

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. Doe...

6 years ago

a) yes b) no c) yes

6 years ago

1. (a) 2. (b) 3. (a) 4. (a)

6 years ago

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...

6 years ago

3*lambda*exp(- 3*t*lambda)

6 years ago

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 expandi...

6 years ago

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,...

6 years ago

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.

6 years ago

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_...

6 years ago