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A non-negative loss in a car accident is distributed with the following probability density function
\[
f(x)=\frac{1}{10} \exp (-x / 10)
\]
for $x \geq 0$. Suppose that first $5$ units of loss is incurred by the insured and the remaining loss if any is covered by the insurance company. What is the expected loss to the insurance company?

  1. $10 \exp \left(-\frac{1}{2}\right)$
  2. $10$
  3. $5 \exp \left(-\frac{1}{2}\right)$
  4. $5+10 \exp \left(-\frac{1}{2}\right)$
  5. $15 \exp \left(-\frac{1}{2}\right)$

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