Probability & Statistics Difficulty: Introductory

Common Probability Distributions

Probability distributions encode the behavior of random variables. The Gaussian, binomial, Poisson, and exponential distributions appear throughout science and machine learning.

Key Points

  • The Gaussian distribution is the maximum-entropy distribution for fixed mean and variance.
  • The exponential distribution is memoryless and models waiting times.
  • The central limit theorem explains the ubiquity of Gaussian distributions.

Formulas

Gaussian PDF
$$f(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$
Binomial PMF
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$
Poisson PMF
$$P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

Code Example

from scipy import stats

X = stats.norm(loc=0, scale=1)
print(X.pdf(0.0))  # 0.3989...
print(X.cdf(1.96))  # ~ 0.975

Tags

  • distributions
  • gaussian

References

  • All of Statistics
    Larry Wasserman · Springer · source
  • Probability and Statistics for Engineers and Scientists
    Ronald E. Walpole et al. · Pearson · source

Knowledge Graph