Let be a random variable with known distribution. Let another random variable be a function of :where . How do we derive the distribution of from the distribution of ?
There is no general answer to this question. However, there are several special cases in which it is easy to derive the distribution of . We discuss these cases below.
When the function is strictly increasing on the support of (i.e. ), then admits an inverse defined on the support of , i.e. a function such thatFurthermore is itself strictly increasing.
The distribution function of a strictly increasing function of a random variable can be computed as follows.
Proposition (distribution of an increasing function) Let be a random variable with support and distribution function . Let be strictly increasing on the support of . Then, the support of isand the distribution function of is
Of course, the support is determined by and by all the values can take. The distribution function of can be derived as follows:
if is lower than than the lowest value can take on, then , so
if belongs to the support of , then can be derived as follows:
if is higher than than the highest value can take on, then , so
Therefore, in the case of an increasing function, knowledge of and of the upper and lower bounds of the support of is all we need to derive the distribution function of from the distribution function of .
Example Let be a random variable with support and distribution functionLetThe function is strictly increasing and it admits an inverse on the support of :The support of is . The distribution function of is
In the cases in which is either discrete or continuous there are specialized formulae for the probability mass and probability density functions, which are reported below.
When is a discrete random variable, the probability mass function of can be computed as follows.
Proposition (probability mass of an increasing function) Let be a discrete random variable with support and probability mass function . Let be strictly increasing on the support of . Then, the support of isand its probability mass function is
This proposition is a trivial consequence of the fact that a strictly increasing function is invertible:
Example Let be a discrete random variable with support and probability mass function Let The support of isThe function is strictly increasing and its inverse isThe probability mass function of is
When is a continuous random variable and is differentiable, then also is continuous and its probability density function can be easily computed as follows.
Proposition (density of an increasing function) Let be a continuous random variable with support and probability density function . Let be strictly increasing and differentiable on the support of . Then, the support of isand its probability density function is
This proposition is a trivial consequence of the fact that the density function is the first derivative of the distribution function: it can be obtained by differentiating the expression for the distribution function found above.
Example Let be a continuous random variable with supportand probability density functionLet The support of isThe function is strictly increasing and its inverse iswith derivativeThe probability density function of is
When the function is strictly decreasing on the support of (i.e. ), then admits an inverse defined on the support of , i.e. a function such thatFurthermore is itself strictly decreasing.
The distribution function of a strictly decreasing function of a random variable can be computed as follows.
Proposition (distribution of a decreasing function) Let be a random variable with support and distribution function . Let be strictly decreasing on the support of . Then, the support of isand the distribution function of is
Of course, the support is determined by and by all the values can take. The distribution function of can be derived as follows:
if is lower than than the lowest value can take on, then , so
if belongs to the support of , then can be derived as follows:
if is higher than than the highest value can take on, then , so
Therefore, also in the case of a decreasing function, knowledge of and of the upper and lower bounds of the support of is all we need to derive the distribution function of from the distribution function of .
Example Let be a random variable with support and distribution functionLetThe function is strictly decreasing and it admits an inverse on the support of :The support of is . The distribution function of iswhere equals when and otherwise (because is always zero except when and ).
We report below the formulae for the special cases in which is either discrete or continuous.
When is a discrete random variable, the probability mass function of can be computed as follows.
Proposition (probability mass of a decreasing function) Let be a discrete random variable with support and probability mass function . Let be strictly decreasing on the support of . Then, the support of isand its probability mass function is
The proof of this proposition is identical to the proof of the proposition for strictly increasing functions. In fact, the only property that matters is that a strictly decreasing function is invertible:
Example Let be a discrete random variable with support and probability mass function Let The support of isThe function is strictly decreasing and its inverse isThe probability mass function of is
When is a continuous random variable and is differentiable, then also is continuous and its probability density function is derived as follows.
Proposition (density of a decreasing function) Let be a continuous random variable with support and probability density function . Let be strictly decreasing and differentiable on the support of . Then, the support of isand its probability density function is
This proposition is easily derived: 1) remembering that the probability that a continuous random variable takes on any specific value is and, as a consequence, for any ; 2) using the fact that the density function is the first derivative of the distribution function; 3) differentiating the expression for the distribution function found above.
Example Let be a uniform random variable on the interval , i.e., a continuous random variable with supportand probability density functionLet where is a constant. The support of iswhere we can safely ignore the fact that , because is a zero-probability event (see Continuous random variables and zero-probability events). The function is strictly decreasing and its inverse iswith derivativeThe probability density function of isTherefore, has an exponential distribution with parameter (see the lecture entitled Exponential distribution).
In the case in which the function is neither strictly increasing nor strictly decreasing, the formulae given in the previous sections for discrete and continuous random variables are still applicable, provided is one-to-one and hence invertible. We report these formulae below.
When is a discrete random variable, the probability mass function of is given by the following.
Proposition (probability mass of a one-to-one function) Let be a discrete random variable with support and probability mass function . Let be one-to-one on the support of . Then, the support of isand its probability mass function is
The proof of this proposition is identical to the proof of the propositions for strictly increasing and strictly decreasing functions found above:
When is a continuous random variable and is differentiable, then also is continuous and its probability density function is given by the following proposition.
Proposition (density of a one-to-one function) Let be a continuous random variable with support and probability density function . Let be one-to-one and differentiable on the support of . Then, the support of isIfthen the probability density function of is
For a proof of this proposition see: Poirier, D. J. (1995) Intermediate statistics and econometrics: a comparative approach, MIT Press.
Below you can find some exercises with explained solutions.
Let be a continuous random variable with supportand probability density functionLet Find the probability density function of .
The support of isThe function is strictly increasing and its inverse iswith derivativeThe probability density function of is
Let be a continuous random variable with supportand probability density functionLet Find the probability density function of .
The support of isThe function is strictly decreasing and its inverse iswith derivativeThe probability density function of is
Let be a discrete random variable with supportand probability mass functionLet Find the probability mass function of .
The support of isThe function is strictly increasing and its inverse isThe probability mass function of is
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