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We are given a stick that extends from 0 to x. Its length, x, is the...

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].

  1. Write down the (fully specified) joint PDF fX,Y(x,y) of X and Y.

    For 0<y≤x:

    fX,Y(x,y)=

  2. 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.

  3. MAP estimate of X:

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