diogenes
Most popular questions and responses by diogenes
math, probability
Problem 1. Determining the type of a lightbulb. The lifetime of a typeA bulb is exponentially distributed with parameter 𝜆 . The lifetime of a typeB bulb is exponentially distributed with parameter 𝜇 , where 𝜇>𝜆>0 . You have a box full of
asked on July 27, 2019 
math, probability
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
asked on August 13, 2019 
fluid mechanics
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, curlyphi). The sphere is immersed in an infinite stream of an ideal
asked on August 13, 2019 
math, probability
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
asked on August 13, 2019 
math, probability
Problem 4. Ships All ships travel at the same speed through a wide canal. Each ship takes t days to traverse the length of the canal. Eastbound ships (i.e., ships travelling east) arrive as a Poisson process with an arrival rate of lambda_E ships per day.
asked on August 13, 2019 
math, probability
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
asked on August 13, 2019 
fluid mechanics
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
asked on August 13, 2019 
fluid mechanics
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 the
asked on August 13, 2019 
math, probability
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
asked on August 13, 2019 
fluid mechanics
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
asked on August 13, 2019

fluid mechanics
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
posted on August 13, 2019 
fluid mechanics
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 zdirection.
posted on August 13, 2019 
fluid mechanics
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,
posted on August 13, 2019 
fluid mechanics
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
posted on August 13, 2019 
math, probability
1. (LW)/(LE+LW) 2. (LE)*e^((LE)*x) 3. e^((LW)*2*t) 4. ((LE)^7*v^6*e^((LE)*v))/(720) 5. 2*LW*LE/((LW+LE)^2) 6.a k1 5.b LE^7*LW^(k7)/(LE+LW)^k
posted on August 13, 2019 
math, probability
3*lambda*exp( 3*t*lambda)
posted on August 13, 2019 
math, probability
1. Since students are equally wellprepared 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
posted on August 13, 2019 
math, probability
1. (a) 2. (b) 3. (a) 4. (a)
posted on August 13, 2019 
math, probability
a) yes b) no c) yes
posted on August 13, 2019 
probability
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
posted on August 13, 2019 
probability
Sorry, (4) is wrong above: Here are my latest answers: let me know if I am wrong. E[YX=x] = 2*x E[Y] = 0 Cov[X,Y]= 2 E[XY=y]= (2*y)/5 Var[XY=y]= 1/5
posted on August 5, 2019 
probability
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)
posted on August 5, 2019 
probability
anyone else getting 0 for the Covariance?
posted on August 5, 2019 
probability
In 2019: 1. ln(mu/(3*lambda))/(mulambda) 2. (a): (1/4)*e^(mu*alpha) + (3/4)(1e^(lambda*alpha)) 3. 0.3286
posted on July 27, 2019 
probability
In 2019 the answer to part 2 is (a). (1/4)*e^(mu*alpha) + (3/4)(1e^(lambda*alpha))
posted on July 27, 2019