Probability: distribution, cumulative distribution

[Robin04]

Homework Statement:

Does a distribution function have an associated cumulative distribution?

Relevant Equations:

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I came across this problem in my assignement but I don't really understand the question. The lectures notes handed out by the teacher does not use the term cumulative distribution. Wikipedia says that a cumulative distribution function is the same as a distribution function.

 

[SammyS]

I came across this problem in my assignment but I don't really understand the question. The lectures notes handed out by the teacher does not use the term cumulative distribution. Wikipedia says that a cumulative distribution function is the same as a distribution function.

As stated on that Wikipedia entry, "cumulative distribution function" (CDF), and "distribution function" generally refer to the same thing.

You may like to refer to the Wikipedia entry for the Probability density function . The third paragraph there mentions some possible confusion with terminology for the probability density function (pdf).

The pdf and the CDF are closely related.

 

Not really sure what your question is, but here is something that might potentially be useful:

Given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a random variable $X: \Omega \to \mathbb{R}$ on this probability space, we have two concepts:

The distribution of $X$, denoted $\mathbb{¨P}_X$, is the function defined by $\mathbb{¨P}_X(A) = \mathbb{P}(X \in A)$, where $A$ is in the Borel-sigma algebra of the real numbers.

Another concept is the distribution function $F_X$ defined by $F_X(x) = \mathbb{P}(X \leq x) = \mathbb{P}(X \in ]-\infty, x]) = \mathbb{P}_X(]-\infty, x])$.

It turns out, as a result of one of the main theorems of measue theory (the uniqueness theorem), that the distribution function determines the distribution of $X$ uniquely.

Hence, distribution function and distribution can be seen as the same thing.

 

[WWGD]

I think issues are those of being Lebesgue integrable, in order to go back and forth.

 

[FactChecker]

I agree with @SammyS . I would interpret them as being the same thing. (Assuming that you did not really mean density versus distribution.)

 

[r0bHadz]

OP are we talking about continuous or discrete random variables?

 

[Ray Vickson]

I came across this problem in my assignement but I don't really understand the question. The lectures notes handed out by the teacher does not use the term cumulative distribution. Wikipedia says that a cumulative distribution function is the same as a distribution function.

Nowadays, the word "cumulative" is often dropped. To speak of non-cumulative probabilities, we have probability mass functions (for the case of a discrete R.V.) or a probability density function (for an (absolutely) continuous R.V.). There can also be mixed cases---partly discrete and partly continuous. (There are also pathological case of continuous random variables that do not have a probability density, but in a lifetime of looking at applications I have never seen one---not to say it can't happen.)

The word "distribution" is also used when describing a probability "law". For example, we may say the something has a geometric distribution or a Poisson distribution or a normal distribution and the like.