Question

Let X be a loss random variable with cdf 0, x<0. The 10th percentile is θ-k. The 90th percentile is 5θ-3k. Determine the value of α. Problem 30.11 Let X be a random variable with density function f(x)-(Wr for x 〉 0 and 0 otherwise. Calculate the mode of X.

Answer 1

10th percentile is $\theta$-k ; i.e F($\theta$-k) = 10%

$\Rightarrow 1-\left ( \frac{\theta}{\theta&plus;\theta-k} \right )^\alpha=0.1 \Rightarrow \left ( \frac{\theta}{2\theta-k} \right )^\alpha=0.9$ ......... [1]

90th% percentile is 5$\theta$-3k ; i.e F(5$\theta$-3k) = 90%

  $\Rightarrow 1-\left ( \frac{\theta}{\theta&plus;5\theta-3k} \right )^\alpha=0.9 \Rightarrow \left ( \frac{\theta}{6\theta-3k} \right )^\alpha=0.1$ .......... [2]

Dividing equation [1] by [2] :-

$\frac{(\frac{\theta}{2\theta-k})^\alpha}{(\frac{\theta}{6\theta-3k})^\alpha}=\frac{0.9}{0.1} \Rightarrow \left ( \frac{6\theta-3k}{2\theta-k} \right )^\alpha=9 \Rightarrow 3^\alpha=3^2 \Rightarrow \alpha =2$