# Why does the PDF use a different variable than x?

In the below image (from Wikipedia but also found in my text book), I noticed that the variable within the integrand is a "u" rather than the "x" which is found in the CDF function. Why is the function being differentiated with respect to u rather than x? It is my understanding that x is the random variable in question due to the subscript X on both the CDF and PDF. • The variable of integration has no significance when evaluating a definite integral: math.stackexchange.com/questions/446591/… – Moss Murderer Aug 15 '18 at 23:45
• @MossMurderer Is there a reason they don't use x there? I'm not understanding why they would use a different variable. – samvoit4 Aug 16 '18 at 0:25
• (1) they could call it $u$, $s$, $y$, it doesn't matter. (2) They must use a different variable than $x$ to integrate over because $x$ is already being used! $x$ is the argument of $F$. – Matthew Gunn Aug 16 '18 at 1:33
• For example, let $f_t$ denote the number of burglaries on day $t$. Then $F_t = \sum_{i=1}^t f_i$ denotes the total number of burglaries that have occurred from day 1 to day $t$. Notice how $F_t = \sum_{t=1}^t f_t$ is completely non-sensical/ambiguous because you're using $t$ in two different ways. – Matthew Gunn Aug 16 '18 at 1:39
• Note that $F$ is a function of $x$ and on the RHS $x$ is already playing its role -- it's the top limit on the integral. So for the same reason that if you are summing up to $t$, you don't use $t$ as the index in the sum (you might sum $i=1,...,t$) - otherwise one variable would be doing two different jobs. The $u$ is a 'dummy' which carries no special meaning, it's just a placeholder like $i$ in a sum. – Glen_b Aug 16 '18 at 1:40

Here, $X$ is a random variable but $x$ is only a number. By definition of the CDF, $$F_X(x) = \mathbb{P}[X \leq x]$$ and to calculate this quantity we need to integrate the PDF of $X$ from $-\infty$ to the value $x$. Since the symbol $x$ is already used for the upper limit of the integral, we need another symbol in the integral. Hence $u$ (or any other symbol) can be used.
If this notation is unfamiliar to you, try this instead $$F_X(a) = \mathbb{P}[X \leq a] = \int_{-\infty}^{a}f_X(x)dx$$