The Poisson distribution

The Poisson distribution models the number of times an event occurs in an interval of space or time. For example, a Poisson random variable \(X\) may be:

• The number earthquakes greater than 6 Richter occurring over the next 100 years.

• The number of major floods over the next 100 years.

• The number of patients arriving at the emergency room during the night shift.

• The number of electrons hitting a detector in a specific time interval.

The Poisson is a good model when the following assumptions are true:

• The number of times an event occurs in an interval takes values \(0,1,2,\dots\).

• Events occur independently.

• The probability that an event occurs is constant per unit of time.

• The average rate at which events occur is constant.

• Events cannot occur at the same time.

When these assumptions are valid, we can write:

\[ X\sim \operatorname{Poisson}(\lambda), \]

where \(\lambda>0\) is the rate with each the events occur. You read this as:

The random variable \(X\) follows a Poisson distribution with rate parameter \(\lambda\).

The PMF of the Poisson is:

\[ p(X=k) = \frac{\lambda^ke^{-\lambda}}{k!}. \]

Let’s look at a specific example. Historical data show that at a given region a major earthquake occurs once every 100 years on average. What is the probability that \(k\) such earthquakes will occur within the next 100 years. Let \(X\) be the random variable corresponding to the number of earthquakes over the next 100 years. Assuming the Poisson model is valid, the rate parameter is \(\lambda = 1\) and we have:

\[ X\sim \operatorname{Poisson}(1). \]

Here is a Poisson random variable:

Show code cell source Hide code cell source

MAKE_BOOK_FIGURES=False

import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import seaborn as sns
sns.set_context("paper")
sns.set_style("ticks")

def set_book_style():
    plt.style.use('seaborn-v0_8-white') 
    sns.set_style("ticks")
    sns.set_palette("deep")

    mpl.rcParams.update({
        # Font settings
        'font.family': 'serif',  # For academic publishing
        'font.size': 8,  # As requested, 10pt font
        'axes.labelsize': 8,
        'axes.titlesize': 8,
        'xtick.labelsize': 7,  # Slightly smaller for better readability
        'ytick.labelsize': 7,
        'legend.fontsize': 7,
        
        # Line and marker settings for consistency
        'axes.linewidth': 0.5,
        'grid.linewidth': 0.5,
        'lines.linewidth': 1.0,
        'lines.markersize': 4,
        
        # Layout to prevent clipped labels
        'figure.constrained_layout.use': True,
        
        # Default DPI (will override when saving)
        'figure.dpi': 600,
        'savefig.dpi': 600,
        
        # Despine - remove top and right spines
        'axes.spines.top': False,
        'axes.spines.right': False,
        
        # Remove legend frame
        'legend.frameon': False,
        
        # Additional trim settings
        'figure.autolayout': True,  # Alternative to constrained_layout
        'savefig.bbox': 'tight',    # Trim when saving
        'savefig.pad_inches': 0.1   # Small padding to ensure nothing gets cut off
    })

def save_for_book(fig, filename, is_vector=True, **kwargs):
    """
    Save a figure with book-optimized settings.
    
    Parameters:
    -----------
    fig : matplotlib figure
        The figure to save
    filename : str
        Filename without extension
    is_vector : bool
        If True, saves as vector at 1000 dpi. If False, saves as raster at 600 dpi.
    **kwargs : dict
        Additional kwargs to pass to savefig
    """    
    # Set appropriate DPI and format based on figure type
    if is_vector:
        dpi = 1000
        ext = '.pdf'
    else:
        dpi = 600
        ext = '.tif'
    
    # Save the figure with book settings
    fig.savefig(f"{filename}{ext}", dpi=dpi, **kwargs)


def make_full_width_fig():
    return plt.subplots(figsize=(4.7, 2.9), constrained_layout=True)

def make_half_width_fig():
    return plt.subplots(figsize=(2.35, 1.45), constrained_layout=True)

if MAKE_BOOK_FIGURES:
    set_book_style()
make_full_width_fig = make_full_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()
make_half_width_fig = make_half_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()

import numpy as np
import scipy.stats as st

X = st.poisson(1.0)

And some samples from it:

X.rvs(10)

array([1, 0, 0, 1, 1, 1, 1, 3, 2, 0])

Let’s plot the PMF:

ks = range(6)
fig, ax = make_full_width_fig()
ax.vlines(ks, 0, X.pmf(ks))
ax.set_xlabel('Number of major earthquakes in next 100 years')
ax.set_ylabel('Probability of occurance')
save_for_book(fig, 'ch9.fig4')

You will investigate the Poisson distribution more in {ref}`lecture09:homework

Questions

• How would the rate parameter \(\lambda\) change if the rate with each major earthquakes occured in the past was 2 every 100 years? Plot the pmf of the new Poisson random variable. You may have to add more points in the x-axis.