problem 1

The PDF is

f (x) = x e^{- x^{2} / 2}

\begin{matrix} P (x \leq a) & = \int_{0}^{a} f (x) dx & (5.1) = \int_{0}^{a} x e^{- x^{2} / 2} dx & (5.2) = 1 - e^{- a^{2} / 2} & (5.3) \end{matrix}

(a)

\begin{matrix} P (1 < X < 3) & = P (X \leq 3) - P (X \leq 1) & (5.4) = e^{- 1 ∕ 2} - e^{- 9 ∕ 2} & (5.5) \end{matrix}

(b) For first quantile $q_{1}$

P (X \leq q_{1}) = \frac{1}{4}

1 - e^{- q_{1}^{2} / 2} = \frac{1}{4}

q_{1} = 0.54

For second quantile $q_{2}$

P (X \leq q_{2}) = \frac{2}{4}

1 - e^{- q_{2}^{2} / 2} = \frac{2}{4}

q_{1} = 0.83

For third quantile $q_{3}$

P (X \leq q_{3}) = \frac{1}{4}

1 - e^{- q_{3}^{2} / 2} = \frac{3}{4}

q_{3} = 1.17