# probability

The vertical coordinate (“height") of an object in free fall is described by an equation of the form

x(t)=θ0+θ1t+θ2t2,
where θ0, θ1, and θ2 are some parameters and t stands for time. At certain times t1,…,tn, we make noisy observations Y1,…,Yn, respectively, of the height of the object. Based on these observations, we would like to estimate the object's vertical trajectory.

We consider the special case where there is only one unknown parameter. We assume that θ0 (the height of the object at time zero) is a known constant. We also assume that θ2 (which is related to the acceleration of the object) is known. We view θ1 as the realized value of a continuous random variable Θ1. The observed height at time ti is Yi=θ0+Θ1ti+θ2t2i+Wi,i=1,…,n, where Wi models the observation noise. We assume that Θ1∼N(0,1), W1,…,Wn∼N(0,σ2), and all these random variables are independent.

In this case, finding the MAP estimate of Θ1 involves the minimization of

θ21+1σ2∑i=1n(yi−θ0−θ1ti−θ2t2i)2
with respect to θ1.

Carry out this minimization and choose the correct formula for the MAP estimate, θ^1, from the options below.

θ^1=∑ni=1ti(yi−θ0−θ2t2i)σ2θ^1=∑ni=1ti(yi−θ0−θ2t2i)σ2+∑ni=1t2iθ^1=∑ni=1ti(yi−θ0−θ2t2i)σ2+∑ni=1θ2t2inone of the above
The formula for θ^1 can be used to define the MAP estimator, Θ^1 (a random variable), as a function of t1,…,tn and the random variables Y1,…,Yn. Identify whether the following statement is true.

The MAP estimator Θ^1 has a normal distribution.

Let σ=1 and consider the special case of only two observations (n=2). Write down a formula for the mean squared error E[(Θ^1−Θ1)2], as a function of t1 and t2. Enter 't1' for t1 and 't2' for t2.

Consider the “experimental design" problem of choosing when to make measurements. Under the assumptions of part (3), and under the constraints 0≤t1,t2≤10, find the values of t1 and t2 that minimize the mean squared error associated with the MAP estimator.

1. 👍
2. 👎
3. 👁
1. 1.Carry out this minimization and choose the correct formula for the MAP estimate, θ^1, from the options below.

(second choice) θ^1=∑ni=1ti(yi−θ0−θ2t2i)σ2+∑ni=1t2i

4. t1 and t2 = 10

1. 👍
2. 👎
2. 2) True

1. 👍
2. 👎
3. 3. ?

1. 👍
2. 👎
4. We can recognize, that our mean squared error = var(Θ), so our final answer will be:
1/(1+t1^2+t2^2)

1. 👍
2. 👎
5. i found now t2 = 10

1. 👍
2. 👎

## Similar Questions

1. ### physics

An object rolls off a tabletop with a horizontal velocity v0x = 2.5 m/s. The table is at a height y0 = 0.55 m, above the floor. Use a coordinate system with its origin on the floor directly beneath the point where the object rolls

2. ### Physics

Calculate the vertical distance an object dropped from rest covers in 12 seconds of free fall. (The unit of measure for distance is the meter, written as m.)

3. ### physics

A 75 kg skydiver in free fall is subjected to a crosswind exerting a force of 60 N and to a vertical air resistance force of 100 N. Calculate the resultant force acting on the skydiver [6 marks] and his angle of fall (relative to

4. ### Science

What would be the velocity of a falling object after 8 seconds of free fall?

1. ### Physics

Two Diamonds begin a free fall from rest from the same height, 1.0 s apart. How long after the first diamond begins to fall will the two diamonds be 10m apart? Please help, I am very confused with this question, I have completed

2. ### MATH

The time (t, in seconds) for a free-falling object to fall d feet is described by the mathematical formula: t=d over the square root of 16 If a worker accidentally drops a hammer from a building and it hit the ground 4 seconds

3. ### math

An object is thrown or fired straight upwards at an initial speed of v_0 ft⁄s will reach height h feet after t seconds, where h and t are related to the formula h=-16t^2+v_0 t Suppose the object is fired straight upwards with an

4. ### physics

On planet x, an object weighs 10.4N. On planet B where the magnitude of the free fall acceleration is 1.48g (where g=9.8 m/s^2 is the gravitational acceleration on Earth),the object weighs 24.6N. The acceleration of gravity is 9.8

1. ### Algebra

Help Please, ty A ball is launched at 20 meters per second (m/s) from a 60 meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 20t + 60, where s is in meters. What is the

2. ### Physics

An object is released from rest at height h above the surface of the Earth, where h is much smaller than the radius of the Earth. It takes t seconds to fall to the ground. At what height should this object be released from rest in

3. ### physics

An object is dropped from a height H above the ground. This free-falling object requires 0.904 s to travel the last 34 m before hitting the ground. From what height H above the ground did the object fall? The acceleration of

4. ### Algebra

The function H(t) = -16t2 + vt + s shows the height H (t), in feet, of a projectile launched vertically from s feet above the ground after t seconds. The initial speed of the projectile is v feet per second. Part A: The projectile