Creating new probability distributions from old ones (or existing ones) is a familiar theme in the study pf probability. This post shows how to generate a distribution under a transformation. The process is illustrated with examples.
Practice problems are found in the next post.
The starting point is the random variable whose probability density function (pdf) is given by the following.
The given is transformed in four different ways as follows:
We demonstrate how to derive the pdfs of these four new random variables based on the pdf given at the beginning. Note that the support of is the interval . Because of the transformations, the supports of the variables are different. The support of is the interval . The support of is also . The support of is . The support of is also .
There are two ways to derive the pdfs of , . One way is the CDF method: to find the CDF of the new variable and then take the derivative to get the pdf. Another way is the method of transformation, which is the focus here. We show how to use CDF method on in order to draw out the idea of the method of transformation.
The and be the pdf and CDF of . Let be the CDF of . The following derives .
Thus the CDF is the CDF evaluated at . Since pdf is the derivative of the CDF, the pdf is obtained by taking derivative of .
The step that is labeled with * is the key step in the derivative and will be discussed in further details.
The Method of Transformation
Let’s describe the method demonstrated in the above derivation. There is a starting probability distribution represented by the random variable . Its pdf is whose support is a subset of the x-axis, likely an interval (of finite or infinite length). Let’s call the support . The support of a pdf is the set of all such that . We have a differentiable function defined on . This function is a one-to-one function over the support . The function does not have to be a one-to-one function over all of the x-axis. It just has to be one-to-one over the support . As a result, the function is either an increasing function or a decreasing function over the support.
Since is a one-to-one function, it has an inverse . The inverse is defined over the set .
Consider the new random variable . The following gives the pdf of .
The formula (1) gives the method of transformation and is illustrated by the step labeled with * above. With , the transformation is the function . It is not a one-to-one function over the entire x-axis but it is a one-to-one function on the support . In fact, is an increasing function over . The inverse function is then . Applying (1) gives the pdf .
One thing to keep in mind is that the method works only if the transformation is a one-to-one function over the support of the original pdf (either is increasing or decreasing). If not, the method will produce a wrong answer. Another thing to keep in mind is that when the transformation is a decreasing function, its inverse is also a decreasing function. Then its derivative would be negative. In (1), we use the absolute value of the derivative.
Using the method of transformation, the following shows the pdfs of , .
It will be instructive to examine the graphs of the pdfs. The following is the graph of the pdf of .
The starting pdf (Figure 1) is a straight line with negative slope. In this distribution, more probabilities are found near zero. For example . About 44% of the values from this distribution are expected to be less than 1. The following is the graph of the pdf of .
Figure 2 shows that the effect of the transformation is to push the probabilities further to zero. The 44% that is less than 1 in Figure 1 is further pushed toward zero. Hence the graph in Figure 2 is extremely positively skewed (or skewed to the right since the right tail is longer). The following is the graph of the pdf of .
Figure 3 shows that the effect of the transformation is to push the probabilities in the opposite direction toward 4. The hence the distribution of is extremely negative skewed (or left skewed since the left tail is longer).
The method of transformation is a great tool making it possible to create new distributions with desired characteristics from old ones.
Some named distributions are generated from transformation. For example, the lognormal distribution is a transformation from the normal distribution where the transformation is an exponential function. More specifically, if has a normal distribution with mean and variance , then has a lognormal distribution and parameters and . The transformation goes the other way too. If has a lognormal distribution and parameters and , then has a normal distribution with mean and variance . See here for a discussion of lognormal distribution. Practice problems on lognormal distribution are found here.
Another named distribution that is generated from a transformation is the Weibull distribution. It is generated by raising an exponential random variable to a power (discussed here). The topic of raising an exponential distribution to a power is further discussed here. For more distributions created by transformation, explore to the site in the given links.
Practice problems are found in the next post.
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