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Probability, Statistics and Financial Data / Multivariate and Derived Distributions

Derived Distributions

A trip duration is 180/(speed)180/(\text{speed}); a log-return is ln⁡(ST/S0)\ln(S_T/S_0); a portfolio P&L is a sum of position P&Ls. Every one of these is a transformed random variable, Y=g(X)Y=g(X) (or Z=X+YZ=X+Y), and finding its distribution — a derived distribution — is one of the most commonly needed operations in probabilistic modelling. This note systematises the approach first previewed for discrete maxima in Cumulative Distribution Functions: compute the CDF of YY, then differentiate. It specialises to a direct formula when gg is strictly monotonic, and closes with convolution, the formula for the PDF of a sum of two independent random variables — the continuous analogue of the discrete convolution from Joint PMFs of Multiple Random Variables.
Learning objectives. By the end of this note you should be able to:
  • apply the two-step CDF method to find the PDF of Y=g(X)Y=g(X) for a general (not necessarily monotonic) function gg;
  • apply the direct formula for the PDF of Y=g(X)Y=g(X) when gg is strictly monotonic and differentiable;
  • state and apply the formula for the PDF of Y=aX+bY=aX+b;
  • derive and apply the convolution formula for the PDF of Z=X+YZ=X+Y, X,YX,Y independent.

In this lesson

  1. The CDF method
  2. The monotonic-function formula
  3. Convolution: the sum of two independent random variables

Before you start

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