CDF and Survival Function¶
The cumulative distribution function (CDF) and survival function (SF) are complementary functions that describe the probability of a random variable falling below or above a threshold.
Cumulative Distribution Function (CDF)¶
The CDF gives the probability that a random variable \(X\) is less than or equal to a value \(x\):
For continuous distributions this is the integral of the PDF from \(-\infty\) to \(x\). For discrete distributions it is the sum of the PMF up to \(k\).
Properties of the CDF¶
The CDF is non-decreasing, right-continuous, and satisfies \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to \infty} F(x) = 1\).
Continuous Example: Normal Distribution CDF¶
import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt
mu = 3.0
a = stats.norm(loc=mu)
x = np.linspace(mu - 3, mu + 3, 100)
y_pdf = a.pdf(x)
y_cdf = a.cdf(x)
plt.plot(x, y_pdf, label='PDF')
plt.plot(x, y_cdf, label='CDF')
plt.legend(loc='lower left')
plt.title(f'Normal Distribution (μ={mu}, σ=1)')
plt.xlabel('x')
plt.show()
The CDF rises from 0 to 1 in an S-shaped (sigmoid) curve. At the mean \(\mu\), the CDF equals 0.5.
Continuous Example: Exponential Distribution CDF¶
The exponential distribution models waiting times and inter-arrival times. Its CDF rises quickly for high rate parameters:
import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt
la = 3.0 # rate parameter λ
a = stats.expon(scale=1/la) # scale = 1/λ
x = np.linspace(0, 3, 100)
y_pdf = a.pdf(x)
y_cdf = a.cdf(x)
plt.plot(x, y_pdf, label='PDF')
plt.plot(x, y_cdf, label='CDF')
plt.legend(loc='lower left')
plt.title(f'Exponential Distribution (λ={la})')
plt.xlabel('x')
plt.show()
Note the important parametrization detail: scipy.stats.expon uses scale = 1/λ, so when the rate is \(\lambda = 3\), you pass scale=1/3.
Discrete Example: Poisson Distribution CDF¶
For discrete distributions, the CDF is a step function. A bar chart effectively shows both the PMF and CDF together:
import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt
mu = 3.0
a = stats.poisson(mu)
x = np.arange(0, 11)
y_cdf = a.cdf(x)
y_pmf = a.pmf(x)
plt.bar(x, y_cdf, label='CDF', alpha=0.5)
plt.bar(x, y_pmf, label='PMF', alpha=0.5)
plt.legend()
plt.title(f'Poisson Distribution (μ={mu})')
plt.xlabel('k')
plt.show()
The CDF bars show the cumulative probability up to each value \(k\), while the PMF bars show the individual probability at each \(k\).
Survival Function (SF)¶
The survival function is the complement of the CDF:
In scipy.stats, use .sf(x) instead of computing 1 - .cdf(x). The dedicated method is numerically more accurate in the tails where CDF values are very close to 1:
a = stats.norm(loc=0, scale=1)
# These are mathematically equivalent, but sf is more accurate in the tails
p1 = 1 - a.cdf(5) # may lose precision
p2 = a.sf(5) # numerically stable
Computing Probabilities Over Intervals¶
The CDF enables computation of interval probabilities:
a = stats.norm(loc=0, scale=1)
prob = a.cdf(1) - a.cdf(-1) # P(-1 < X ≤ 1) ≈ 0.6827
Summary¶
The CDF and survival function are essential tools for probability computation. In scipy.stats, .cdf() gives \(P(X \le x)\) and .sf() gives \(P(X > x)\) with numerical stability. Together with the PDF/PMF and quantile functions, they form the complete interface for working with probability distributions.