In this problem, we study a simple noisy communication channel. Suppose that X is a binary signal that takes value −1 and 1 with equal probability. This signal X is sent through a noisy communication channel, and the medium of transmission adds an independent noise term. More precisely, the received signal is Y=X+N, where N is standard normal, independent of X.
The decoder receives the value y of Y, and decides whether X was 1 or −1, using the following decoding rule: it decides in favor of 1 if and only if
P(X=1|Y=y)>2P(X=0|Y=y).
It turns out that the decoding rule can be expressed in the form: decide in favor of 1 if and only if Y>t, for some threshold t. Find the threshhold t.
As an intermediate step, find p1≜P(X=1|Y=y).
p1= unanswered
Now find t.
t= unanswered
p1 = 1/(1 + e^(-2*y)) <-- answer should be in terms of y not x
t?
I would try solving P(X=1|Y=t)>2P(X=0|Y=t)
t>(ln2)/2
1. (1/2)*e^((2*y-1)/2) I don't understand the other answer given here, but I've got this
2. ln(2)/2
To find the threshold t, we first need to compute p1, which represents the probability that X = 1 given Y = y.
To find p1, we can use Bayes' theorem:
P(X=1|Y=y) = (P(Y=y|X=1) * P(X=1)) / P(Y=y)
Since X is equally likely to be -1 or 1, we have P(X=1) = P(X=-1) = 1/2.
The distribution of Y given X can be written as:
Y = X + N, where N is standard normal (mean = 0, variance = 1) and independent of X.
Therefore, P(Y=y|X=1) = P(X+N=y|X=1) = P(N=y-1) = ϕ(y-1), where ϕ(x) is the standard normal probability density function.
Similarly, P(Y=y|X=-1) = P(X+N=y|X=-1) = P(N=y+1) = ϕ(y+1).
Now, we can compute p1:
p1 = P(X=1|Y=y) = (P(Y=y|X=1) * P(X=1)) / P(Y=y)
= (ϕ(y-1) * 1/2) / (ϕ(y-1) * 1/2 + ϕ(y+1) * 1/2)
= ϕ(y-1) / (ϕ(y-1) + ϕ(y+1))
After computing p1, we can find the threshold t such that P(X=1|Y=y) > 2P(X=-1|Y=y), which can be rewritten as p1 > 2(1-p1).
To find the value of t, we need to solve the equation p1 = 2(1-p1) for p1. Simplifying the equation, we get:
p1 = 2 - 2p1
3p1 = 2
p1 = 2/3
Now that we know p1, we can find t by solving the equation p1 = ϕ(t-1) / (ϕ(t-1) + ϕ(t+1)) for t.
Since it is not possible to solve this equation analytically, we can use numerical methods or approximations to find the solution for t. One possible approach is to estimate t using a numerical solver or approximation techniques such as Newton's method.
Therefore, the values of p1 and t cannot be determined exactly without additional calculations or approximations.