its considered bad form to write
because it can get confusing when "x" appears in the limits of the integral and the integral itself.
Instead they have changed the name of the variable in the integral from x to t.
Hi, I have a problem of understanding the definition of cumulative distribution function for a continuous variable.
Standard definition in the textbook as well as in many internet sites is:
F(x) = (integration from minus infinity to x) f(t)dt. (sorry i do not know how to type mathematical symbol here, please see Cumulative distribution function - Wikipedia, the free encyclopedia).
What does t mean in the definition of cumulative distribution for a continuous variable? why not use f(x)dx?
Thank you in advance.
No, the if x is a limit of integration it is not the variable of integration. The variable of integration is a dummy any symbol can be used for it, while the limit of integration is not a dummy variable.
You should never see:
because you are now using the same symbol to mean two different things in the same expression.
CB
I think you have things a little backwards. The cumulative distribution function (CDF) of a random variable is defined by
A PDF for , denoted is (essentially) defined by
A function which satisfies this requirement may or may not exist, but the CDF is always going to exist, so it wouldn't make sense to, in general, define the CDF in terms of a PDF. Of course, a PDF completely defines a random variable so you can define random variables by just giving a PDF if one exists.