Question

Let Θ be an unknown random variable that we wish to estimate. It has a prior distribution with mean 1 and variance 2. Let W be a noise term, another unknown random variable with mean 3 and variance 5. Assume that Θ and W are independent.

We have two different instruments that we can use to measure Θ. The first instrument yields a measurement of the form X1=Θ+W, and the second instrument yields a measurement of the form X2=2Θ+3W. We pick an instrument at random, with each instrument having probability 1/2 of being chosen. Assume that this choice of instrument is independent of everything else. Let X be the measurement that we observe, without knowing which instrument was used.

Give numerical answers for all parts below.

E[X]=

- unanswered

E[X2]=

- unanswered

The LLMS estimator of Θ given X is of the form aX+b. Give the numerical values of a and b.

a=

- unanswered


 

b=

- unanswered

Answer 1

1. 7.5
2. 97
3. 0.071
4. 0.467

Answer 2

Can someone explain me how to find E[X]? thanks in advance