admits at most a countable number of roots (i.e., a finite, or countably infinite, number of such that ) then the previous relation between the probability density functions can be generalized with where , according to the inverse function theorem. The formulas for densities do not demand to be increasing. In the measure-theoretic, axiomatic approach to probability, if a random variable on and a Borel measurable function , then is also a random variable on , since the composition of measurable functions is also measurable. (However, this is not necessarily true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of . Example 1 Let be a real-valued, continuous random variable and let . If , then , so If , then so Example 2 Suppose is a random variable with a cumulative distribution where is a fixed parameter. Consider the random variable Then, The last expression can be calculated in terms of the cumulative distribution of so which is the cumulative distribution function (CDF) of an exponential distribution. Example 3 Suppose is a random variable with a standard normal distribution, whose density is Consider the random variable We can find the density using the above formula for a change of variables: In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). However, because of symmetry, both halves will transform identically, i.e., The inverse transformation is and its derivative is Then, This is a chi-squared distribution with one degree of freedom. Example 4 Suppose is a random variable with a normal distribution, whose density is Consider the random variable We can find the density using the above formula for a change of variables: In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms: The inverse transformation is and its derivative is Then, This is a noncentral chi-squared distribution with one degree of freedom. Some properties The probability distribution of the sum of two independent random variables is the convolution of each of their distributions. Probability distributions are not a vector space—they are not closed under linear combinations, as these do not