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Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given a known average rate. It is commonly used in scenarios like traffic flow, call arrivals, or biological processes.

Key Topics

Generating Poisson Distribution

NumPy's random.poisson() function simulates Poisson-distributed random events. You specify the average rate of events (lambda) and the size of the dataset.

Example

# Generating Poisson distribution
import numpy as np

# Average rate (lambda) = 3, Size = 1000
calls = np.random.poisson(lam=3, size=1000)

print("First 10 call counts:", calls[:10])

Output

First 10 call counts: [3 1 4 2 3 0 3 2 5 1]

Explanation: The random.poisson() function generates random numbers based on the Poisson distribution. Here, the average rate of events is 3, representing the expected number of calls per interval.

Visualizing the Distribution

You can use a histogram to visualize the Poisson distribution and analyze the frequency of events within intervals.

Example

# Visualizing Poisson distribution
import seaborn as sns
import matplotlib.pyplot as plt

# Data
calls = np.random.poisson(lam=3, size=1000)

# Plot
sns.histplot(calls, kde=False, color="orange", bins=15)
plt.title("Poisson Distribution of Call Arrivals")
plt.xlabel("Number of Calls")
plt.ylabel("Frequency")
plt.show()

Output

A histogram showing the frequency of call arrivals modeled using a Poisson distribution.

Explanation: The histogram shows the frequency distribution of call counts, demonstrating the typical shape of a Poisson distribution.

Key Takeaways

  • Event-Based Modeling: The Poisson distribution models the number of events in a fixed interval with a known average rate.
  • Simulation: Use random.poisson() to simulate scenarios like traffic flow, customer arrivals, or biological processes.
  • Visualization: Histograms help understand the frequency and patterns of events.
  • Applications: Analyze phone calls, website hits, or rare event occurrences.
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