Homework 11: The Cumulative Distribution Function

1. Section 3.3, #2.
2. Let \(X\) be the lifetime of a certain electronic device (measured in hours). Suppose that the probability density function of \(X\) is given by $$ f_X(x)= \begin{cases} \frac{10}{x^2}, & x \gt 10\\ 0, & x\leq 10 \end{cases} $$
   1. Find and sketch \(F_X(x)\).
   2. Find \(P(X \gt 20)\) using the cdf.
3. A continuous random variable \(X\) has pdf given by $$ f_X(x)= \begin{cases} 1 - |x|, &|x|\leq 1\\ 0, & |x| \gt 1 \end{cases} $$
   1. Find and sketch \(F_X(x)\).
   2. Find \(P(X \lt 1.5)\) using the cdf.
4. A continuous random variable \(X\) has cdf given by $$ F_X(x)= \begin{cases} 0, & x\leq 0\\ \frac{1}{9}x^2, & 0 \lt x\leq 3\\ 1, & x\gt 3 \end{cases} $$
   1. Find and sketch \(f_X(x)\).
   2. Find \(P(1 \lt X\leq 2)\) using the cdf.