Suppose that the random pair (y,x) has a joint distribution on {0,1}x(0,infinity) that may be specified as follows:
P[y=1]=.7 and P[y=0]=.3
conditional on the value of y, the variable x is Exponential with mean y+1
A "density" for (y,x) (that one adds over x and integrates over y) is then
f(y,x)= .7I[y=0]exp(-x)+.3I[y=1](.5*exp(-x/2)) if (y,x) is in {0,1}x(0,infinity)
0 otherwise
a) Evaluate P[x>1]
b) Find the conditional probability that y=1 given the value of x, P[y=1|x]
c) Find the 0-1 loss optimal predictor of y based on x
d) Evaluate the risk of your predictor in c)