# Normal Distribution

> ***GitHub***: https://github.com/czs108

## Definition

The *Normal Distribution* is in the shape of a bell curve. The curve is *symmetrical*, with the *highest* probability density in the *center* of the curve. The probability density *decreases* when getting away from the *mean*. Both the *mean* and *median* are at the center and have the highest probability density.

The *Normal Distribution* is defined by **2** parameters. $\mu$ is where the center of the curve is, and $\sigma^{2}$ is the spread. If a continuous random variable $X$ follows a *Normal Distribution* with *mean* $\mu$ and *standard deviation* $\sigma$, this is generally written as:

\begin{equation}
X \sim N(\mu,\, \sigma^{2})
\end{equation}

As $\sigma^{2}$ gets *larger*, the *flatter* and *wider* the normal curve becomes.

## Expectation

\begin{equation}
E(X) = \mu
\end{equation}

## Variance

\begin{equation}
Var(X) = \sigma^{2}
\end{equation}

## Calculating

### Standardization

\begin{equation}
Z = \frac{X - \mu}{\sigma}
\end{equation}

### Probability Table

Start off by calculating $z$ to **2** decimal places, then read off the corresponding probability $P(Z < z)$.

The *1st* *column* gives the value of $z$ to **1** decimal place (without rounding), and the top *row* gives the *2nd* decimal place. The probability is where the two intersect.

| z       | .00    | .01    | .02    | .03    | .04    | ...  |
| ------- | ------ | ------ | ------ | ------ | ------ | ---- |
| **0.0** | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | ...  |
| **0.1** | 0.5398 | 0.5438 | 0.5478 | 0.5517 | 0.5557 | ...  |
| ...     | ...    | ...    | ...    | ...    | ...    | ...  |

For example, if you wanted to find $P(Z < 0.12)$, you'd find **0.1** in the *1st* column, **.02** in the top row, and read off a probability of **0.5478**.

## Approximation

Suppose $q$ is $(1 - p)$, a *Binomial Distribution* $X \sim B(n,\, p)$ can be approximated by $X \sim N(np,\, npq)$ normally if $np > 5$ and $nq > 5$.

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Jupyter Notebook
exercises/Normal Distribution.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
exercises/Normal Distribution.ipynb
czs108/Probability-Theory-Exercises
60c6546db1e7f075b311d1e59b0afc3a13d93229
MIT
exercises/Normal Distribution.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