binomial distribution added

contrib/machine-learning/binomial_distribution.md

@@ -0,0 +1,123 @@
+# 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):
+
+\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]
+
+where:
+- \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).
+
+## Properties of Binomial Distribution
+
+- **Mean:** \( \mu = np \)
+- **Variance:** \( \sigma^2 = np(1 - p) \)
+- **Standard Deviation:** \( \sigma = \sqrt{np(1 - p)} \)
+
+## 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
+
+1. **Import necessary libraries:**
+
+    ```python
+    import numpy as np
+    import matplotlib.pyplot as plt
+    from scipy.stats import binom
+    ```
+
+2. **Define parameters:**
+
+    ```python
+    # Number of trials
+    n = 10
+    # Probability of success
+    p = 0.5
+    # Number of successes
+    k = np.arange(0, n + 1)
+    ```
+
+3. **Compute the PMF:**
+
+    ```python
+    pmf = binom.pmf(k, n, p)
+    ```
+
+4. **Plot the PMF:**
+
+    ```python
+    plt.bar(k, pmf, color='blue')
+    plt.xlabel('Number of Successes')
+    plt.ylabel('Probability')
+    plt.title('Binomial Distribution PMF')
+    plt.show()
+    ```
+
+5. **Compute the CDF:**
+
+    ```python
+    cdf = binom.cdf(k, n, p)
+    ```
+
+6. **Plot the CDF:**
+
+    ```python
+    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:
+
+```python
+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()

contrib/machine-learning/index.md

@@ -1,3 +1,4 @@
 # List of sections
 
 - [Section title](filename.md)
+- [Binomial Distribution](binomial_distribution.md)