Exponential Distribution

📈 Exponential Distribution in Python

The Exponential Distribution models the time between independent events that occur at a constant rate.
It is closely related to the Poisson Distribution, which models the number of events in a fixed interval.

1. Characteristics of Exponential Distribution

Continuous distribution
Parameter: scale = 1/λ (λ = event rate)
Probability Density Function (PDF):

$\frac{1}{\text{scale}} e^{-x/\text{scale}} \quad \text{for } x \ge 0$

Mean = scale
Standard Deviation = scale

Applications:

Time between customer arrivals
Lifespan of electronic components
Waiting times in queues

2. Generate Exponential Data Using NumPy

Output (example):

Values ≥ 0
Skewed distribution

3. Visualize Exponential Distribution

Histogram is right-skewed
KDE shows the smooth probability density

4. Change Scale Parameter

data_small = np.random.exponential(scale=1.0, size=1000)
data_large = np.random.exponential(scale=5.0, size=1000)sns.histplot(data_small, bins=30, color='red', label='scale=1', kde=True, alpha=0.5)
sns.histplot(data_large, bins=30, color='blue', label='scale=5', kde=True, alpha=0.5)
plt.title("Exponential Distribution Comparison")
plt.legend()
plt.show()

data_small = np.random.exponential(scale=1.0, size=1000)

data_large = np.random.exponential(scale=5.0, size=1000)sns.histplot(data_small, bins=30, color='red', label='scale=1', kde=True, alpha=0.5)

sns.histplot(data_large, bins=30, color='blue', label='scale=5', kde=True, alpha=0.5)

plt.title("Exponential Distribution Comparison")

plt.legend()

plt.show()

Smaller scale → more concentrated near 0
Larger scale → more spread out

5. Mean and Standard Deviation

Both should approximately equal the scale parameter

🧠 Summary Table

Function	Parameters	Description
`np.random.exponential()`	scale, size	Generate exponential random numbers
`scale`	Mean / 1/λ	Average time between events
`size`	Number of samples	Output array size

🎯 Practice Exercises

Simulate waiting times between calls with scale=3 for 1000 samples.
Compare distributions with scale=1, 2, 5 in one plot.
Compute mean and standard deviation of generated data and compare with scale.

Exponential Distribution