# Poisson Distribution - Broken Machine

> This document is written in *R*.
>
> ***GitHub***: https://github.com/czs108

## Background

> Suppose the *mean* number of machine malfunctions per week, or rate of malfunctions, is **3.4**.

## Question A

> What's the probability of the machine *not* malfunctioning next week?

\begin{equation}
\lambda = 3.4
\end{equation}

\begin{equation}
\begin{split}
P(X = 0) &= \frac{e^{-3.4} \times {3.4}^{0}}{0!} \\
    &= e^{-3.4} \\
    &= 0.033
\end{split}
\end{equation}

Use the `dpois` function directly.


```R
dpois(x=0, lambda=3.4)
```


0.0333732699603261


Or use the `exp` function.


```R
exp(-3.4)
```


0.0333732699603261


## Question B

> What's the probability of the machine malfunctioning **3** times next week?

\begin{equation}
\begin{split}
P(X = 3) &= \frac{e^{-3.4} \times {3.4}^{3}}{3!} \\
    &= 0.2186
\end{split}
\end{equation}


```R
dpois(x=3, lambda=3.4)
```


0.218617167086776


## Question C

> What's the *expectation* and *variance* of the machine malfunctions?

\begin{equation}
E(X) = \lambda = 3.4
\end{equation}

\begin{equation}
Var(X) = \lambda = 3.4
\end{equation}

7589b30aea0d77a794a11456bb372cfb7a654e2b
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Jupyter Notebook
exercises/Poisson Distribution - Broken Machine.ipynb
czs108/Probability-Theory-Exercises
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exercises/Poisson Distribution - Broken Machine.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
exercises/Poisson Distribution - Broken Machine.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
2022-03-21T05:04:07.000Z
2022-03-21T05:04:07.000Z
Qwen/Qwen-72B
1. YES
2. YES