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Binomial Distribution

Introduction

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is commonly used in statistics and probability theory.

Key Characteristics

Number of trials (n): The number of independent experiments or trials.
Probability of success (p): The probability of success on an individual trial.
Number of successes (k): The number of successful outcomes in n trials.

The binomial distribution is defined by the probability mass function (PMF):

P(X = k) = (n choose k) p^k (1 - p)^(n - k)

where:

(n choose k) is the binomial coefficient, calculated as n! / (k!(n-k)!).

Properties of Binomial Distribution

Mean: μ = np
Variance: σ² = np(1 - p)
Standard Deviation: σ = √(np(1 - p))

Python Implementation

Let's implement the binomial distribution using Python. We'll use the scipy.stats library to compute the binomial PMF and CDF, and matplotlib to visualize it.

Step-by-Step Implementation

Import necessary libraries:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

Define parameters:

# Number of trials
n = 10
# Probability of success
p = 0.5
# Number of successes
k = np.arange(0, n + 1)

Compute the PMF:
```
pmf = binom.pmf(k, n, p)
```

Plot the PMF:

plt.bar(k, pmf, color='blue')
plt.xlabel('Number of Successes')
plt.ylabel('Probability')
plt.title('Binomial Distribution PMF')
plt.show()

Compute the CDF:
```
cdf = binom.cdf(k, n, p)
```

Plot the CDF:

plt.plot(k, cdf, marker='o', linestyle='--', color='blue')
plt.xlabel('Number of Successes')
plt.ylabel('Cumulative Probability')
plt.title('Binomial Distribution CDF')
plt.grid(True)
plt.show()

Complete Code

Here is the complete code for the binomial distribution implementation:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

# Parameters
n = 10  # Number of trials
p = 0.5  # Probability of success

# Number of successes
k = np.arange(0, n + 1)

# Compute PMF
pmf = binom.pmf(k, n, p)

# Plot PMF
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.bar(k, pmf, color='blue')
plt.xlabel('Number of Successes')
plt.ylabel('Probability')
plt.title('Binomial Distribution PMF')

# Compute CDF
cdf = binom.cdf(k, n, p)

# Plot CDF
plt.subplot(1, 2, 2)
plt.plot(k, cdf, marker='o', linestyle='--', color='blue')
plt.xlabel('Number of Successes')
plt.ylabel('Cumulative Probability')
plt.title('Binomial Distribution CDF')
plt.grid(True)

plt.tight_layout()
plt.show()

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