What Does P(X ∈ dx) Mean in Probability Notation?

[gnob]

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.

 

[I like Serena]

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]

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.

 

[I like Serena]

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.

 

[blue_raver22]

Hello,

The notation $P(X \in dx)$ or $P(X \in [x, x+dx])$ represents the probability of a continuous random variable $X$ falling within a very small interval $dx$ around a particular value $x$. This notation is commonly used in probability and statistics to represent the probability distribution of a continuous random variable.

In your examples, the first notation $P(Z_v \in dy)$ represents the probability of a random variable $Z_v$ falling within a small interval $dy$, where $Z_v$ follows a Gamma distribution with shape parameter $v$. The second notation $P\left( \int_0^{\tau} e^{\sigma B_s - p\sigma^2 s/2} ds \in du,\,\, B_{\tau} \in dy\right)$ represents the joint probability of a random variable $B_{\tau}$ falling within a small interval $dy$ and the integral of a stochastic process $B_s$ falling within a small interval $du$.

To better understand these notations, I would recommend looking into textbooks on probability and statistics, such as "Introduction to Probability" by Joseph K. Blitzstein and Jessica Hwang, or "Mathematical Statistics and Data Analysis" by John A. Rice. These books provide a rigorous explanation of probability notations and concepts. Additionally, online resources such as Khan Academy and Coursera offer free courses on probability and statistics that can also help clarify these notations.

I hope this helps and good luck in your studies!