Translating informal probability statements to formal counterparts: Difference between revisions
(Created page with "(inspired by how Aluffi explains universal properties in ''Algebra: Chapter 0'') I think probability is one of the fields in math where people who have learned the subject of...") |
No edit summary |
||
| Line 10: | Line 10: | ||
| Let <math>X_1, X_2, X_3, \ldots</math> be i.i.d. || || | | Let <math>X_1, X_2, X_3, \ldots</math> be i.i.d. || || | ||
|- | |- | ||
| <math>\mathbf E_{z \sim \mathcal D}[f(z)]</math> || || | | <math>\mathbf E_{z \sim \mathcal D}[f(z)]</math> || again, here we're giving a random variable <math>z</math> without specifying the sample space, and then we're taking its expected value. the reason why this makes sense is that the expected value does not does not depend on the sample space, as long as the distribution is the same. || | ||
|- | |- | ||
| rolling one die, and then another die || the confusion here is that we're supposed to have a sample space, but here it seems like we're modifying the sample space. again the informal-ness comes from the fact that we have not fully specified the sample space. here we can think of starting with one sample space, and then "growing" it, or we could just start with the bigger sample space from the start || | | rolling one die, and then another die || the confusion here is that we're supposed to have a sample space, but here it seems like we're modifying the sample space. again the informal-ness comes from the fact that we have not fully specified the sample space. here we can think of starting with one sample space, and then "growing" it, or we could just start with the bigger sample space from the start || | ||
|} | |} | ||
Revision as of 06:33, 10 August 2019
(inspired by how Aluffi explains universal properties in Algebra: Chapter 0)
I think probability is one of the fields in math where people who have learned the subject often make many "informal" statements, which can be easily formalized given enough experience, but which are difficult to fully understand if one does not have that experience. Unfortunately, the formal details are often left unexplained. On this page, we give some examples of these "informal" statements, with some thoughts on how to go about translating them to precise mathematical statements.
| Informal statement | Why the statement is informal | Formal statement |
|---|---|---|
| Let be a normally distributed random variable. | A random variable is a function , but here we have not specified the sample space | |
| Let be i.i.d. | ||
| again, here we're giving a random variable without specifying the sample space, and then we're taking its expected value. the reason why this makes sense is that the expected value does not does not depend on the sample space, as long as the distribution is the same. | ||
| rolling one die, and then another die | the confusion here is that we're supposed to have a sample space, but here it seems like we're modifying the sample space. again the informal-ness comes from the fact that we have not fully specified the sample space. here we can think of starting with one sample space, and then "growing" it, or we could just start with the bigger sample space from the start |
![{\displaystyle \mathbf {E} _{z\sim {\mathcal {D}}}[f(z)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5dc74dded88b6e37c185459a15de45c0e14208)
