Question

The density of \(X\) is given by $$ f(x)=\left\\{\begin{array}{ll} 10 / x^{2}, & \text { for } x>10 \\ 0, & \text { for } x \leq 10 \end{array}\right. $$ What is the distribution of \(X ?\) Find \(P[X>20\\}\).

Short answer

The distribution of \(X\) is given by the cumulative distribution function (CDF) \(F(x) = \left\\{ \begin{array}{ll} 0, & \text{for } x\leq 10 \\ 10\left(\frac{1}{10} - \frac{1}{x}\right), & \text{for } x > 10 \end{array}\right.\). The probability \(P[X > 20]\) is 0.5 (50%).

Step-by-step solution

Identify the density function

In this exercise, the density function is given as$$ f(x) = \left\\{ \begin{array}{ll} 10 / x^{2}, & \text{for } x > 10 \\ 0, & \text{for } x \leq 10 \end{array} \right. $$

Find the cumulative distribution function (CDF)

The cumulative distribution function (CDF), denoted by F(x), is the integral of the density function f(x) with respect to x. Let's compute the CDF for x > 10. $$ F(x) = \int_{10}^{x} 10 / t^2 dt = 10\int_{10}^{x} t^{-2} dt $$ Now we can integrate with respect to t: $$ F(x) = 10\left[-\frac{1}{t}\right]_{10}^{x} = 10 \left(\frac{1}{10} - \frac{1}{x}\right) $$ Finally, let's write the CDF for all values of x: $$ F(x) = \left\\{ \begin{array}{ll} 0, & \text{for } x\leq 10 \\ 10\left(\frac{1}{10} - \frac{1}{x}\right), & \text{for } x > 10 \end{array} \right. $$

Calculate the probability P[X > 20]

To find the probability P[X > 20], we can use the CDF as follows: $$ P[X > 20] = 1 - F(20) $$ Now we can substitute the value of F(20) from the CDF formula: $$ P[X > 20] = 1 - 10\left(\frac{1}{10} - \frac{1}{20}\right) = 1 - \frac{1}{2} = \frac{1}{2} $$ So, the probability P[X > 20] is 0.5 (50%).