Probability
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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?  ?
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