2.6. Transformation of univariate random variables
One of the practical benefits of the random variable (RV) concept is the unification of the treatment of random phenomena. In the context of considering transformations of random phenomena for example, the RV concept turns out to be extremely convenient. One might for example be interested in the results of transforming the measurement of a random phenomenon into a different (arbitrary) scale. An example of direct relevance for seismic hazard analysis (in particular the measurement of ground motion amplitudes) is the change between a linear and logarithmic scale. Expressed in terms of the random variable X, this corresponds to the calculation of a new random variable Y which can be derived from X by applying a function u, in other words Y=u(X). In certain cases, one might be interested only in particular statistics of the new random variable, e. g. the mean value or the variance, which simplifies the problem. In other cases, the calculation of the full distribution is of interest. We start the discussion of random variable transformations with a simple linear case.
.............. 2.6. Transformation of univariate random variables
.............. 2.6.1. Linear change-of-units
.............. 2.6.2. Expectation and variance under transformation
.............. 2.6.3. Distributions
.............. 2.6.4. General change-of-variable transformation