A point is chosen uniformly at random from a disk of radius 1, centered at the...

A point is chosen uniformly at random from a disk of radius 1, centered at the origin. Let R be the distance of the point from the origin, and Θ the angle, measured in radians, counterclockwise with respect to the x-axis, of the line connecting the origin to the point.

1. Find the joint distribution function of (R,Θ); i.e. find F(r,θ) = P(R ≤ r, Θ ≤ θ).

2. Are R and Θ independent? Explain your answer.

Expert Solution

milcah answered 1 year ago

Break a stick of unit length at a uniformly chosen random point. Then take the shorted...

Break a stick of unit length at a uniformly chosen random point. Then take the shorted of the two pieces and break it again in two pieces at a uniformly chosen random point. Let X denote the shortest of the final three pieces. Find the density of X.

6. Pick a uniformly chosen random point (X, Y ) inside a unit square [0, 1]×[0,...

6. Pick a uniformly chosen random point (X, Y ) inside a unit square [0, 1]×[0, 1], and let M = min(X, Y ). Find the probability that M < 0.3.

a) Let D be the disk of radius 4 in the xy-plane centered at the origin....

a) Let D be the disk of radius 4 in the xy-plane centered at the origin. Find the biggest and the smallest values of the function f(x, y) = x 2 + y 2 + 2x − 4y on D. b) Let R be the triangle in the xy-plane with vertices at (0, 0),(10, 0) and (0, 20) (R includes the sides as well as the inside of the triangle). Find the biggest and the smallest values of the function...

A horizontal disk with radius R and charge density σ is centered originally. A particle with...

A horizontal disk with radius R and charge density σ is centered originally. A particle with charge Q and mass m is in equilibrium at a height h above the center of the plate, as shown. A gravitational field is present. R = 1.5 m σ = -5⋅10-7 C / m2 h = 2 m Q = -2 µC a) Calculate the mass m of the particle if it is at equilibrium (use g = 9.81 m / s2 )....

1.) Build the parametric equations of a circle centered at the point (2,-3) with a radius...

1.) Build the parametric equations of a circle centered at the point (2,-3) with a radius of 5 that goes counterclockwise and t=0 gives the location (7,-3) 2.) Build the parametric equations for an ellipse centered at the point (2, -3) where the major axis is parallel to the x-axis and vertices at (7, -3) and (-3, -3), endpoints of the minor axis are (2, 0) and (2, -6). The rotation is counterclockwise 3.) Build the parametric equations for a...

A uniformly charged non-conducting sphere of radius 12 cm is centered at x=0. The sphere is...

A uniformly charged non-conducting sphere of radius 12 cm is centered at x=0. The sphere is uniformly charged with a charge density of ρ=+15 μC/m3. Find the work done by an external force when a point charge of +20 nC that is brought from infinity on the x-axis at a distance of 1 cm outside the surface of the sphere. Given the point charge held at its final position, what is the net electric field at x=5 cm on the...

A uniformly charged disk of radius 35.0 cm carries a charge density of 6.40 ✕ 10-3...

A uniformly charged disk of radius 35.0 cm carries a charge density of 6.40 ✕ 10-3 C/m2. Calculate the electric field on the axis of the disk at the following distances from the center of the disk. (a) 5.00 cm MN/C (b) 10.0 cm MN/C (c) 50.0 cm MN/C (d) 200 cm MN/C

A 8.0-cm-diameter disk emits light uniformly from its surface. 20 cm from this disk, along its...

A 8.0-cm-diameter disk emits light uniformly from its surface. 20 cm from this disk, along its axis, is an 9.0-cm-diameter opaque black disk; the faces of the two disks are parallel. 20 cm beyond the black disk is a white viewing screen. The lighted disk illuminates the screen, but there�s a shadow in the center due to the black disk. What is the diameter of the completely dark part of this shadow?

A thin, uniform disk with mass M and radius R is pivoted about a fixed point...

A thin, uniform disk with mass M and radius R is pivoted about a fixed point a distance x from the center of mass. The disk rotates around the fixed point, with the axis of rotation perpendicular to the plane of the disk. What is the period as a function of x, M, R, and acceleration due to gravity (g), and what is the distance x that gives the smallest period?

An arrow lands uniformly random on a circular target of radius r. Let y and z...

An arrow lands uniformly random on a circular target of radius r. Let y and z be the Cartesian coordinates. Thus, we can view y to be a realization of a random variable Y, and z a realization of another random variable Z. (a) Are Y and Z independent? (b) Are Y and Z uncorrelated?

Question