14. Suppose that the cumulative distribution function of the random variab is given by

$$F(x) = 1-e^{-x^2}$$

x>0 Evaluate

(a) P(X > 2);

(b) P(1 < X < 3);

(c) the hazard rate funk of F;

(d) E[X];

(e) Var(X).

HINT: For parts

(d) and

(e) you might want to make use of the resul Theoretical Exercise 5.

15. The number of years that a washing machine functions is a random vari whose hazard rate function is given by

$$\lambda(t) =

\begin{cases}

2 & 0

2+3(t-2) & 2 \le t <5 \\

1.1 & t \ge 5

\end{cases}

$$

(a) What is the probability that the machine will still be working six y after being purchased?

(b) If it is still working six years after being purchased, what is the conditi probability that it will fail within the succeeding two years?