Suppose the \(p d f\) of the continuous random variable \(X\) is \(f(x)=1\), for \(0

a. Draw a sketch of the \(p d f\). Verify that the area under the \(p d f\) for \(0

b. Find the \(c d f\) of \(X\). 

c. Compute the probability that \(X\) falls in each of the intervals \([0,0.1],[0.5,0.6]\), and \([0.79,0.89]\). Indicate the probabilities on the sketch drawn in (a).

d. Find the expected value of \(X\).

e. Show that the variance of \(X\) is \(1 / 12\).

f. Let \(Y\) be a discrete random variable taking the values 1 and 0 with conditional probabilities \(P(Y=1 \mid X=x)=x\) and \(P(Y=0 \mid X=x)=1-x\). Use the Law of Iterated Expectations to find \(E(Y)\).

g. Use the variance decomposition to find \(\operatorname{var}(Y)\).