Suppose the daily revenue (in thousands of dollars) of a small exports firm is uniformly distributed with minimum -1 and maximum 3. Suppose also that the daily cost of operation is 0.1 plus 20% of the squared revenue. What is the cumulative distribution function of the cost?
So I got C=0.1+0.2$R^{2}$ as the cost.
$$ F_{c}(C)=P(C\le c) = P(0.1+0.2R^{2} \le c) $$ $$ P(0.1+0.2R^{2} \le c) = P(-\sqrt{5c-0.5} \le R \le \sqrt{5c-0.5}) $$ $$ P(-\sqrt{5c-0.5} \le R \le \sqrt{5c-0.5}) = P(R \le \sqrt{5c-0.5})-P(R \le -\sqrt{5c-0.5})$$
So I found the bounds from solving $\sqrt{5c-0.5}$ = 3 and $-\sqrt{5c-0.5}$=-1. The bounds are 0.1, 0.3 and 1.9.
The CDF I have so far is: $$F_{c}(C)=\begin{cases} 0,\text{if}\ c\le 0.1, \\ ? ,\text{if}\ 0.1<c\le 0.3, \\ ?, \text{if}\ 0.3<c\le1.9,\\ 1, \text{if}\ c > 1.9, \end{cases}$$.
I am not sure how to find the functions for the leftover cases. I have solved R as $\frac{1}{4}$ from the uniform distribution formula $\frac{1}{\beta - \alpha}$. I evaluated $\frac{1}{4}$ from $-\sqrt{5c-0.5} to \sqrt{5c-0.5}$ and got $\frac{\sqrt{5c-0.5}}{2}$, but not sure how this fits into the CDF. Any help is appreciated. Thanks!
As you noticed, if $c > 0.1$ \begin{equation} F_C(c) = P(-\sqrt{5c-0.5} \leq R \leq \sqrt{5c-0.5})=F_R(\sqrt{5c-0.5})-F_R(-\sqrt{5c-0.5}), \end{equation} where $F_R$ is the CDF of $R$. You may use an explicit form of $F_R$ to finalize this solution. The uniform CDF on $[-1,3]$ writes: \begin{equation} F_R(r) = \delta(r>3) + \frac{(r+1)}{4}\delta(-1 \leq r\leq3), \end{equation} where $\delta$ is the indicator function.