[Term Entry] Python:SciPy scipy.stats: probability distributions

content/scipy/concepts/scipy-stats/terms/probability-distributions/probability-distributions.md

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+---
+Title: 'Probability Distributions'
+Description: 'In SciPy, a probability distribution defines the likelihood of outcomes for a random variable, with functions for density, cumulative probability, and sampling.'
+Subjects:
+  - 'Computer Science'
+  - 'Data Science'
+Tags:
+  - 'Data'
+  - 'Functions'
+  - 'Math'
+  - 'Python'
+CatalogContent:
+  - 'learn-python-3'
+  - 'paths/computer-science'
+---
+
+In statistics and probability theory, a probability distribution describes how the values of a random variable are spread or distributed. It gives the probabilities of the possible outcomes of an experiment or event. In the context of SciPy, the `scipy.stats` module provides a wide range of probability distributions that can be used for modelling, simulating, and analyzing random processes.
+
+SciPy's `scipy.stats` module includes continuous and discrete distributions. **Continuous distributions**, such as normal or exponential distributions, are used to model variables that can take any real value within a range. **Discrete distributions**, such as the binomial or Poisson distributions, model scenarios where outcomes are limited to specific, countable values.
+
+The `scipy.stats` module offers various functions for each distribution type, including:
+
+- **Probability Density Function (PDF)**: Describes the likelihood of a given value under a continuous distribution.
+- **Cumulative Distribution Function (CDF)**: Gives the probability that a random variable will take a value less than or equal to a specified value.
+- **Random Variates**: Functions to generate random samples from a specified distribution.
+- **Statistical Moments**: Functions for calculating the mean, variance, skewness, and kurtosis of the distribution.
+
+These distributions are essential tools for tasks such as hypothesis testing, statistical modelling, simulations, and generating synthetic data that follows known statistical properties.
+
+## Syntax
+
+Each distribution is represented by a corresponding class in `scipy.stats`, which provides methods for computing properties like probability density, cumulative distribution, and random sampling.
+
+```pseudo
+from scipy import stats
+
+# For continuous distributions
+dist = stats.norm(loc=0, scale=1)  # Normal distribution with mean 0 and std 1
+
+# For discrete distributions
+dist_binom = stats.binom(n=10, p=0.5)  # Binomial distribution with 10 trials and 0.5 probability of success
+```
+
+### Methods available for probability distributions
+
+- `pdf(x)`: Probability Density Function (for continuous distributions)
+- `cdf(x)`: Cumulative Distribution Function
+- `rvs(size)`: Random variates (sampling from the distribution)
+- `mean()`: Mean of the distribution
+- `std()`: Standard deviation of the distribution
+
+## Examples
+
+### Normal Distribution
+
+In this example, the probability density function (PDF) will be calculated, and random samples from a normal distribution will be generated.
+
+```py
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy import stats
+
+# Define a normal distribution with mean 0 and standard deviation 1
+dist = stats.norm(loc=0, scale=1)
+
+# Generate 1000 random samples from the distribution
+samples = dist.rvs(size=1000)
+
+# Plot the histogram of the samples
+plt.hist(samples, bins=30, density=True, alpha=0.6, color='g')
+
+# Plot the PDF of the normal distribution
+x = np.linspace(-4, 4, 100)
+pdf = dist.pdf(x)
+plt.plot(x, pdf, 'k', linewidth=2)
+plt.title('Normal Distribution: Mean = 0, Std = 1')
+plt.show()
+```
+
+This code generates random samples from a standard normal distribution and visualizes both the histogram of the samples and the probability density function (PDF).
+
+The output will be:
+
+![Normal Distribution](https://raw.githubusercontent.com/Codecademy/docs/main/media/normal-distribution.png)
+
+## Binomial Distribution
+
+In this example, the binomial distribution will be calculated to compute the probability of a specific number of successes in a series of trials.
+
+```py
+from scipy import stats
+
+# Define a binomial distribution with 10 trials and probability of success 0.5
+dist_binom = stats.binom(n=10, p=0.5)
+
+# Probability of getting exactly 5 successes
+prob_5_successes = dist_binom.pmf(5)
+print(f"Probability of 5 successes: {prob_5_successes}")
+
+# Generate 1000 random samples
+samples_binom = dist_binom.rvs(size=1000)
+
+# Plot the histogram of the binomial samples
+import matplotlib.pyplot as plt
+plt.hist(samples_binom, bins=10, density=True, alpha=0.6, color='blue')
+plt.title('Binomial Distribution: n = 10, p = 0.5')
+plt.show()
+```
+
+In this case, a binomial distribution with 10 trials and a 50% chance of success is defined. The example computes the probability of obtaining exactly 5 successes and visualizes the distribution of random samples.
+
+The output will be:
+
+```shell
+Probability of 5 successes: 0.2460937500000002
+```
+
+![Binomial Distribution](https://raw.githubusercontent.com/Codecademy/docs/main/media/binomial-distribution.png)