Question: 1)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

1)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.)

What is the (unconditional) PDF fX(x) of X?

For 0x1:

a) fX(x)= ?

Calculate E[X].

b) E[X]= ?

2) 2 points possible (graded, results hidden)

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

Find P(TailsX=1/4).

P(TailsX=1/4)=

unanswered

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

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