# Thread: I have a problem of understanding the definition of cumulative distribution function

1. ## I have a problem of understanding the definition of cumulative distribution function

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?

2. its considered bad form to write

$\displaystyle \intop^{x}_{-\infty}f(x) dx$

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.

$\displaystyle \intop^{x}_{-\infty}f(t) dt$

3. Great! I understand it now! So t here is essentially x but for the purpose of clarity we use t instead of x!

Thank you so much, I really really appreciate that XD

4. Originally Posted by Real9999
Great! I understand it now! So t here is essentially x but for the purpose of clarity we use t instead of x!

Thank you so much, I really really appreciate that XD
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:

$\displaystyle \int_{-\infty}^x f(x)\;dx$

because you are now using the same symbol to mean two different things in the same expression.

CB

5. Thank you so much XD

6. Originally Posted by Real9999
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).
I think you have things a little backwards. The cumulative distribution function (CDF) of a random variable $X$ is defined by

$\displaystyle F(x) = P(X \le x).$

A PDF for $X$, denoted $f$ is (essentially) defined by

$\displaystyle F(x) = \int_{-\infty} ^ x f(t) \ dt.$

A function $f$ 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.

7. Thank you very much XD I appreciate all your helps!