# Calculating Normal CDF Probabilities and Plotting the Distribution in Python

## How to Calculate Normal CDF Probabilities in Python

Python is a powerful programming language widely used in data analysis and scientific computing. One of the most common tasks in these fields is calculating probabilities under the normal distribution.

The normal distribution is a continuous probability distribution that often arises in real-world problems, including measurements of physical and social phenomena. It has many applications in fields such as finance, engineering, and medicine, among others.

In this article, we will explain how to calculate normal cumulative distribution function (CDF) probabilities using Python. We will also show you how to plot the CDF using matplotlib.pyplot.

### Using the norm.cdf() Function

The `norm.cdf()` function is part of the `scipy.stats` module, which provides statistical functions in Python. This function calculates the area under the standard normal distribution curve to the left of a given value.

For example, suppose we want to find the probability that a random variable X, normally distributed with mean 0 and standard deviation 1, is less than or equal to 1. We can use `norm.cdf()` as follows:

``````from scipy.stats import norm
p = norm.cdf(1)
``````

The variable `p` now contains the probability that X is less than or equal to 1, which is approximately 0.8413.

Note that `norm.cdf()` takes the value of X as an argument and returns the area under the curve to the left of X.

### Finding the Probability of X Being Less Than a Given Value

To find the probability that X is less than a given value `x`, we can use the same `norm.cdf()` function and pass `x` as the argument. For example, let’s find the probability that X is less than -2:

``````p = norm.cdf(-2)
``````

The variable `p` now contains the probability that X is less than -2, which is approximately 0.02275.

### Finding the Probability of X Being Greater Than a Given Value

To find the probability that X is greater than a given value `x`, we can subtract the probability of X being less than or equal to `x` from 1. For example, let’s find the probability that X is greater than 1:

``````p = 1 - norm.cdf(1)
``````

The variable `p` now contains the probability that X is greater than 1, which is approximately 0.1587.

## Plotting Normal CDF in Python

Visualizing the normal distribution is often useful to understand the behavior of random variables. We can use matplotlib.pyplot to plot the CDF of a normal distribution.

### Importing Necessary Libraries

First, we need to import the necessary libraries. We will use numpy to generate the x-values and scipy.stats to calculate the y-values of the CDF. We will also use matplotlib.pyplot to create the plot.

``````import numpy as np
import scipy.stats as ss
import matplotlib.pyplot as plt
``````

### Defining x and y Values for CDF Plot

We can define x-values using numpy’s linspace function, which returns evenly spaced numbers over a specified interval. We will create a range of x-values from the first percentile to the 99th percentile with 1000 data points.

``````x = np.linspace(ss.norm.ppf(0.01), ss.norm.ppf(0.99), 1000)
``````

We then calculate the y-values of the CDF using the `ss.norm.cdf()` function:

``````y = ss.norm.cdf(x)
``````

### Plotting CDF Using Defined Values

We can now plot the CDF using matplotlib.pyplot’s `plot()` function:

``````plt.plot(x, y)
``````

We should see a curve that resembles the standard normal distribution.

### Modifying Color and Axis Labels for Plot

Finally, we can modify the color and add labels to the plot. For example, we can make the plot blue and add a title, xlabel, and ylabel:

``````plt.plot(x, y, color='blue')
plt.title('Standard Normal Distribution CDF')
plt.xlabel('x')
plt.ylabel('Cumulative Probability')
``````

## Conclusion

In this article, we have shown how to calculate normal cumulative distribution function (CDF) probabilities using Python. We have also demonstrated how to plot the CDF using matplotlib.pyplot.

These techniques are useful for understanding the behavior of random variables under the normal distribution, which has many applications in statistics and science. We hope that this article has been informative and helpful for your data analysis projects.

This article outlined how to calculate normal cumulative distribution function (CDF) probabilities and plot the CDF using Python. By using the `norm.cdf()` function from the `scipy.stats` module, we can easily find the probability of a random variable being less than or greater than a given value.

Visualizing the normal distribution through plotting the CDF is also essential in understanding the behavior of random variables, which is useful in several fields, including statistics and science. Learning how to calculate normal CDF probabilities and plot the CDF using Python is crucial for data analysis projects.