Multinomial Distribution

🎲 Multinomial Distribution in Python

The Multinomial Distribution generalizes the binomial distribution to more than two outcomes per trial.
It models the number of occurrences of each outcome in a fixed number of independent trials.

1. Characteristics of Multinomial Distribution

n → number of trials
pvals → list of probabilities for each category (sum must = 1)
size → number of experiments / samples
Returns counts of outcomes for each category

Example Use Cases:

Rolling a dice multiple times
Drawing colored balls from a bag
Simulating categorical outcomes

2. Generate Multinomial Data Using NumPy

import numpy as np

# Parameters
n = 10 # number of trials
pvals = [0.2, 0.3, 0.5] # probabilities of each category
size = 5 # number of experiments

# Generate multinomial random data
data = np.random.multinomial(n, pvals, size)

print(data)

import numpy as np

# Parameters

n = 10 # number of trials

pvals = [0.2, 0.3, 0.5] # probabilities of each category

size = 5 # number of experiments

# Generate multinomial random data

data = np.random.multinomial(n, pvals, size)

print(data)

Output (example):

Each row → one experiment of n trials
Each column → counts of a category
Sum of each row = n

3. Visualize Multinomial Distribution

Example: 3 categories

import matplotlib.pyplot as plt

# Sum of counts per category across experiments
category_counts = data.sum(axis=0)
categories = ['A', 'B', 'C']

plt.bar(categories, category_counts, color=['red','green','blue'])
plt.title('Multinomial Distribution Counts')
plt.xlabel('Category')
plt.ylabel('Total Counts')
plt.show()

import matplotlib.pyplot as plt

# Sum of counts per category across experiments

category_counts = data.sum(axis=0)

categories = ['A', 'B', 'C']

plt.bar(categories, category_counts, color=['red','green','blue'])

plt.title('Multinomial Distribution Counts')

plt.xlabel('Category')

plt.ylabel('Total Counts')

plt.show()

Shows total occurrences per category across all experiments

4. Change Probabilities or Trials

Probabilities and number of trials directly influence the counts
Sum of counts per row always = n

5. Compare Multiple Experiments

Average counts approach n * pvals as number of experiments increases

🧠 Summary Table

Function	Parameters	Description
`np.random.multinomial()`	n, pvals, size	Generates counts per category in multinomial experiments
`n`	Number of trials	Total trials per experiment
`pvals`	List of probabilities	Must sum to 1
`size`	Number of experiments	Rows in the output array

🎯 Practice Exercises

Roll a 6-sided dice 10 times, repeat 1000 experiments, and plot average counts.
Simulate drawing balls from a bag with 3 colors (probabilities [0.1,0.3,0.6]) for 20 draws and 5 experiments.
Verify that the sum of counts per row always equals n.

Multinomial Distribution