Solution to Exercise 4.10.

  (a) The figure below represents my crude attempt to sketch the graph.

(b)

$$\int_{- \infty}^{\infty} \, f(x;k,\theta) \, dx \; = \; \i... ... \; = \; k \theta^k \int_{\theta}^{\infty} \, x^{-(k+1)} \, dx$$

$$\; = \; k \theta^k \left[ \frac{x^{-(k+1)+1}}{-(k+1)+1} \ri... ...} \; = \; \theta^k \left[ -(\infty)^{-k} + \theta^{-k} \right]$$

$$\; = \; \theta^k \left[ 0 + \theta^{-k} \right] \; = \; \theta^k \theta^{-k} \; = \; 1 .$$

(c)

$$P[X \le b] \; = \; \int_{- \infty}^{b} \, f(x;k,\theta) \, dx \; = \; \int_{\theta}^{b} \, \frac{k \theta^k}{x^{k+1}} \, dx$$

$$\; = \; \theta^k \left[ -b^{-k} + \theta^{-k} \right] \; = \; 1 - (\theta/b)^k .$$

(d)

$$P[a \le X \le b] \; = \; P[ X \le b ] - P[X < a ] \; = \; P[ X \le b ] - P[X \le a ]$$

$$\; = \; 1 - (\theta/b)^k \, - \, \left[ 1 - (\theta/a)^k \right] \; = \; \theta^k \left[ 1/a^k - 1/b^k \right]$$