Probability distributions

• How to Find a PMF and CDF
  Learn how to find a probability mass function and cumulative distribution function from a discrete distribution, with a worked example.
• What Is the Expected Maximum of Two Exponential Variables?
  For independent exponential variables with rates λ₁ and λ₂, the expected maximum is 1/λ₁ + 1/λ₂ − 1/(λ₁ + λ₂).
• How to Find the Expected Value of an Exponential Function
  Find E[e^X] by integrating against X's distribution. For beta and gamma random variables, standard formulas give the result and its conditions.
• What Is the Difference Between the Rician and Gaussian Distributions?
  A Rician variable is the magnitude of a Gaussian vector with a possible nonzero mean; compare its support, density, limiting cases, and uses.
• Conditional Mean and Variance of an Exponential Random Variable
  For an exponential random variable conditioned to exceed a threshold, memorylessness gives a shifted exponential: mean threshold plus 1/rate and variance 1/rate squared.
• How Is the Multivariate Gaussian Density Derived?
  Derive the multivariate Gaussian density by transforming a standard normal vector, and see how the covariance determinant and inverse enter the formula.
• How to Generate Random Numbers From a Probability Equation
  To sample from a probability equation, define a valid distribution, then use inverse-CDF sampling or rejection sampling when inversion is difficult.
• How to Find a Probability Density Function (PDF)
  Learn how to derive a probability density function from a CDF, identify a known distribution, or transform a random variable, then check its support and integral.
• What Is the MGF of the Negative Binomial Distribution?
  For the failures-before-rth-success convention, the negative binomial MGF is [p/(1−(1−p)e^t)]^r; its domain and other conventions are explained.
• How to Prove the Variance of a Geometric Distribution
  Derive Var(X) = (1-p)/p² for a geometric random variable using its probability-generating function, and see how the 0-based convention differs.
• What Is the Standard Deviation of an Exponential Distribution?
  For an exponential distribution with rate λ, the standard deviation is 1/λ; with scale θ or mean μ, it equals θ or μ. See the derivation and example.