# Solution {-}

A random variable $X$ whose probability density function is given as:

\begin{equation*}
  f_X(x)=
  \begin{cases}
    \alpha e^{-\alpha x}, &x \geq 0 \\
    0, &x < 0 \\
  \end{cases}
\end{equation*}

This density function is used to describe the failure of equipment components. The probability that a component will fail within time $T$ is:

\begin{equation*}
  P(fail)=\int_0^T f_X(x) dx
\end{equation*}

Find $\alpha$ for an electronic component whose expected lifetime is 10000 hours:

\begin{equation*}
  E(X)=\int_0^\infty x \cdot \alpha e^{-\alpha x} dx=\alpha\frac{1}{\alpha^2}=\frac{1}{\alpha}=10000 \rightarrow \alpha=\frac{1}{10000}
\end{equation*}


```python
from sympy import exp, integrate, symbols, oo

x, alpha = symbols('x alpha') 

EX = integrate(x*alpha*exp(-alpha*x), (x, 0, oo))

EX
```




$\displaystyle \begin{cases} \frac{1}{\alpha} & \text{for}\: \left|{\arg{\left(\alpha \right)}}\right| < \frac{\pi}{2} \\\int\limits_{0}^{\infty} \alpha x e^{- \alpha x}\, dx & \text{otherwise} \end{cases}$



a4300da2bc12b3cd38d8290e86eaa7db9adbb4b5
ipynb
Jupyter Notebook
Problem 1.15.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-07T09:36:36.000Z
2021-09-07T09:36:36.000Z
Problem 1.15.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
Problem 1.15.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-20T18:48:20.000Z
2021-09-20T18:48:20.000Z
Qwen/Qwen-72B
1. YES
2. YES