In: Statistics and Probability
Let X has the probability density function (pdf)
f(x)={C1, if 0 < x ≤ 1,
C2x, if1<x≤4,
0, otherwise.
Assume that the mean E(X) = 2.57.
(a) Find the normalizing constants C1 and C2.
(b) Find the cdf of X, FX.
(c) Find the variance Var(X) and the 0.28 quantile q0.28 of X.
(d)LetY =kX. Find all constants k such that Pr(1<Y <9)=0.035. Hint: express the event {1 < Y < 9} in terms of the random variable X and then use the cdf of X, FX.