11.3 Exercise 10

Truncated exponential. Suppose $X \sim$ exponential $\left(\lambda\right)$ and $Y = I_{\left[0 , a\right]} \left(X\right) X + I_{\left(a , \infty\right)} \left(X\right) a$.

1. Use the fact that
   $\int_{0}^{\infty} t e^{- \lambda t} d t = \frac{1}{\lambda^{2}} \text{and} \int_{a}^{\infty} t e^{- \lambda t} d t = \frac{1}{\lambda^{2}} e^{- \lambda a} \left(1 + \lambda a\right)$
   to determine an expression for $E \left[Y\right]$.
2. Use the approximation method, with $\lambda = 1 / 50 , a = 30$. Approximate the exponential at 10,000 points for $0 \leq t \leq 1000$. Compare the approximate result with the theoretical result of part (a).