A random variable X is generated as follows. We flip a coin. With probability p, the result is Heads, and then X is generated according to a PDF fX|H which is uniform on [0,1]. With probability 1p the result is Tails, and then X is generated according to a PDF fX|T of the form

fX|T(x)=2x,if x[0,1].

(The PDF is zero everywhere else.)

1. What is the (unconditional) PDF fX(x) of X?
2. For 0x1:
3. fX(x)=
4. Calculate E[X].
5. E[X]=

We now wish to estimate the result of the coin toss, based on the value of X.

1. Find P(TailsX=1/4).
2. P(TailsX=1/4)=

The MAP rule decides in favor of Heads if Xa. What is a?

1. a=