Application in Probability
Substitution can be used to answer the following important question in probability: given a random variable with probability density and another random variable related to by the equation, what is the probability density for ?
It is easiest to answer this question by first answering a slightly different question: what is the probability that takes a value in some particular subset ? Denote this probability . Of course, if has probability density then the answer is
but this isn't really useful because we don't know py; it's what we're trying to find in the first place. We can make progress by considering the problem in the variable . takes a value in S whenever X takes a value in, so
Changing from variable x to y gives
Combining this with our first equation gives
so
In the case where and depend on several uncorrelated variables, i.e., and, can be found by substitution in several variables discussed above. The result is
Read more about this topic: Integration By Substitution
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