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On the Notation P(X ϵ dx)

gnob

New member
Dec 11, 2012
11
Good day!
I was trying to make sense of the notation $P(X \in dx),$ where $X$ is a continuous random variable. Some also write this one as $P(X \in [x, x+dx])$ to represent the probability that the random variable $X$ takes on values in the interval $[x, x+dx].$

I have seen similar notation a lot in my readings such as the ff:
$$
P(Z_v \in dy) = \frac{1}{\Gamma(v)}e^{-y}y^{v-1} dy \quad\quad \text{and}\quad\quad
P\left( \int_0^{\tau} e^{\sigma B_s - p\sigma^2 s/2} ds \in du,\,\, B_{\tau} \in dy\right).
$$

Please help me understand these notations, or better yet please suggest me a (reference) book that rigorously explains these notations?
Thanks in advance.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,856
Good day!
I was trying to make sense of the notation $P(X \in dx),$ where $X$ is a continuous random variable. Some also write this one as $P(X \in [x, x+dx])$ to represent the probability that the random variable $X$ takes on values in the interval $[x, x+dx].$

I have seen similar notation a lot in my readings such as the ff:
$$
P(Z_v \in dy) = \frac{1}{\Gamma(v)}e^{-y}y^{v-1} dy \quad\quad \text{and}\quad\quad
P\left( \int_0^{\tau} e^{\sigma B_s - p\sigma^2 s/2} ds \in du,\,\, B_{\tau} \in dy\right).
$$

Please help me understand these notations, or better yet please suggest me a (reference) book that rigorously explains these notations?
Thanks in advance.
Welcome to MHB, gnob! :)

As far as I'm concerned $P(X \in dx)$ is bad notation.
According to your explanation it refers to an $x$, but that $x$ is not part of the notation, which is bad.

A more usual notation is $P(X = x)dx$ which is the same as $P(x \le X < x + dx)$.
We're talking about a probability density here, which is the probability that an event occurs in a (very) small interval.
Usually, you'd use a probability density to define a probability between 2 boundaries.
Like:
$$P(a \le X < b) = \int_a^b P(X=x)dx$$
 

gnob

New member
Dec 11, 2012
11
Welcome to MHB, gnob! :)

As far as I'm concerned $P(X \in dx)$ is bad notation.
According to your explanation it refers to an $x$, but that $x$ is not part of the notation, which is bad.

A more usual notation is $P(X = x)dx$ which is the same as $P(x \le X < x + dx)$.
We're talking about a probability density here, which is the probability that an event occurs in a (very) small interval.
Usually, you'd use a probability density to define a probability between 2 boundaries.
Like:
$$P(a \le X < b) = \int_a^b P(X=x)dx$$
Thanks for your time and reply. Its really of great help.
Though I want to ask if you know of some books, that discusses the above topic.
Thanks a lot.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,856
Thanks for your time and reply. Its really of great help.
Though I want to ask if you know of some books, that discusses the above topic.
Thanks a lot.
Hmm, books?
Well, I guess that would be any book where probability is explained in combination with calculus.
Sorry, but I don't have any book in particular in mind.