probability distributions

• 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 the geometric distribution’s variance, q/p², using its probability-generating function, and compare the trials and failures counting conventions.
• 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.