Probability
posted by A .
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]= ?
E[X2]= ?
The LLMS estimator of Θ given X is of the form aX+b. Give the numerical values of a and b.
a= ?
b= ?

Probability 
Anonymus
E[X]=7.5

Probability 
Anonymous
E[X^2] = 97

Probability 
qwerty
a=?
b=? 
Probability 
anonymous
a and b ?

Probability 
Mary
a and b ?????

Probability 
anonymous
How did you get 97?

Probability 
Anonym
If anybody knows how to get the E[X^2] can you enlighten us please? I am just really thrown off by the fact that it has 2 ways of measuring X.

Probability 
Anonym
E[X]=7.5
E[X^2]=98.5
a=0.071
b=0.467 
Probability 
Anon
re: E[X^2]:
Remember that Var[X] = E[X^2](E[X])^2 and also that (in shorthand) Var[X] = E[Var] + Var[E]. Use this last identity to get Var[X]; we already have E[X] so we can easily get E[X^2]. 
Probability 
Anonymous
correct answer for E[X^2] is 98.5
(NOT 97) 
Probability 
Sam
b=0.471

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

Probability 
Anonymous
you can find E[X] using the law of total expectation
Respond to this Question
Similar Questions

stats
Consider an infinite population with 25% of the elements having the value 1, 25% the value 2, 25% the value 3, and 25% the value 4. If X is the value of a randomly selected item, then X is a discrete random variable whose possible … 
probability
A random experiment of tossing a die twice is performed. Random variable X on this sample space is defined to be the sum of two numbers turning up on the toss. Find the discrete probability distribution for the random variable X and … 
probability April005
X is a random variable following binomial distribution with mean 2.4 and variance 1.44 find 
Probability
7. The random variable X is distributed normally with a mean of 12.46 and variance of 13.11. You collect a random sample of size 37. a. What is the probability that your sample mean is between 12 and 13? 
Math/Probability
The random variable X has a lognormal distribution, when the mean of ln(X) = 5.45 and variance of ln(X) = 0.334, what is the probability that X >139.76? 
probability
A fair coin is flipped independently until the first Heads is observed. Let K be the number of Tails observed before the first Heads (note that K is a random variable). For k=0,1,2,…,K, let Xk be a continuous random variable that … 
Probability
Consider a fire alarm that senses the environment constantly to figure out if there is smoke in the air and hence to conclude whether there is a fire or not. Consider a simple model for this phenomenon. Let Θ be the unknown true … 
Statistics/probability
The random variable X has a binomial distribution with the probability of a success being 0.2 and the number of independent trials is 15. The random variable xbar is the mean of a random sample of 100 values of X. Find P(xbar<3.15). 
Probability
Consider a fire alarm that senses the environment constantly to figure out if there is smoke in the air and hence to conclude whether there is a fire or not. Consider a simple model for this phenomenon. Let Θ be the unknown true … 
Probability
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 …