Probability
posted by A .
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 state of the environment: Θ=1 means that there is a fire and Θ=0 means that there is no fire. The signal observed by the alarm at time n is Xn=Θ+Wn, where the random variable Wn represents noise. Assume that Wn is Gaussian with mean 0 and variance 1 and is independent of Θ. Furthermore, assume that for i≠j, Wi and Wj are independent. Suppose that Θ is 1 with probability 0.1 and 0 with probability 0.9.
Give numerical answers for all parts below.
Given the observation X1=0.5, calculate the posterior distribution of Θ. That is, find the conditional distribution of Θ given X1=0.5.
P(Θ=0∣X1=0.5)= ?
P(Θ=1∣X1=0.5)= ?
What is the LMS estimate of Θ given X1=0.5?
θˆLMS= ?
What is the resulting conditional mean squared error of the LMS estimator given X1=0.5?  ?
Respond to this Question
Similar Questions

Math!
From the top of a 40m fire tower, a fire ranger observes smoke in tower locations. One has a an angle of depression of 10o, and the other has an angle of depression of 7o. Calculate the distance between the smoke sightings when they … 
science
here are some science questions I've been trying to do for over an hour and i just cant figure them out! 1.how does a fan convert energy? 
Trianle Fire
What was the significance of the fire which occurred on March 25, 1911 at the Triangle Shirtwaist Company in New York City? 
Trigonometry (Law of Sine)
Fire towers A and B are located 10 miles apart. They use the direction of the other tower as 0°. Rangers at fire tower A spots a fire at 42°, and rangers at fire tower B spot the same fire at 64°. How far from tower A is the fire … 
Trigonometry (Law of Sine)
Fire towers A and B are located 10 miles apart. They use the direction of the other tower as 0°. Rangers at fire tower A spots a fire at 42°, and rangers at fire tower B spot the same fire at 64°. How far from tower A is the fire … 
Math
Q. In a decrement timed drill, at the sound of a fire alarm it took a fire fighter 9.5 seconds to reach the Fire pole, 3.2 seconds to slide down the pole, 13.6 to put on protective gear, 5.7 to climb aboard the truck. How long did … 
Trig application
I worked this problem out but I do not fully understand what I am doing. Fire tower A is 30 kilometers due west of fire tower B.. A fire is spotted from the towers, and the bearings from A and B are N 76 degrees E and N 56 degrees … 
English
1. What do we have to do in case of a fire? 
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 … 
English
1. He built a fire in/on the mountain. 2. He made a fire in/on the mountain. 3. He started a fire in/on the mountain. 4. He lighted a fire in/on the mountain.  Can we use all the verbs?