We are given a stick that extends from 0 to x. Its length, x, is the realization of an exponential random variable X, with mean 1. We break that stick at a point Y that is uniformly distributed over the interval [0,x].
Write down the (fully specified) joint PDF fX,Y(x,y) of X and Y.
For 0<y≤x:
fX,Y(x,y)=
Find Var(E[Y∣X]).
Var(E[Y∣X])=
3. We do not observe the value of X, but are told that Y=2.2. Find the MAP estimate of X based on Y=2.2.
MAP estimate of X:
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