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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-orvc | Structured version Visualization version GIF version |
Description: Define the preimage set
mapping operator. In probability theory, the
notation 𝑃(𝑋 = 𝐴) denotes the probability that a
random variable
𝑋 takes the value 𝐴. We
introduce here an operator which
enables to write this in Metamath as (𝑃‘(𝑋∘_{RV/𝑐} I 𝐴)), and
keep a similar notation. Because with this notation (𝑋∘_{RV/𝑐} I 𝐴)
is a set, we can also apply it to conditional probabilities, like in
(𝑃‘(𝑋∘_{RV/𝑐} I 𝐴) ∣ (𝑌∘_{RV/𝑐} I 𝐵))).
The oRVC operator transforms a relation 𝑅 into an operation taking a random variable 𝑋 and a constant 𝐶, and returning the preimage through 𝑋 of the equivalence class of 𝐶. The most commonly used relations are: - equality: {𝑋 = 𝐴} as (𝑋∘_{RV/𝑐} I 𝐴) cf. ideq 5506- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘_{RV/𝑐} E 𝐴) cf. epel 5257- less-than: {𝑋 ≤ 𝐴} as (𝑋∘_{RV/𝑐} ≤ 𝐴) Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g., for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.) |
Ref | Expression |
---|---|
df-orvc | ⊢ ∘_{RV/𝑐}𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^{◡}𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cR | . . 3 class 𝑅 | |
2 | 1 | corvc 31062 | . 2 class ∘_{RV/𝑐}𝑅 |
3 | vx | . . 3 setvar 𝑥 | |
4 | va | . . 3 setvar 𝑎 | |
5 | 3 | cv 1657 | . . . . 5 class 𝑥 |
6 | 5 | wfun 6116 | . . . 4 wff Fun 𝑥 |
7 | 6, 3 | cab 2810 | . . 3 class {𝑥 ∣ Fun 𝑥} |
8 | cvv 3413 | . . 3 class V | |
9 | 5 | ccnv 5340 | . . . 4 class ^{◡}𝑥 |
10 | vy | . . . . . . 7 setvar 𝑦 | |
11 | 10 | cv 1657 | . . . . . 6 class 𝑦 |
12 | 4 | cv 1657 | . . . . . 6 class 𝑎 |
13 | 11, 12, 1 | wbr 4872 | . . . . 5 wff 𝑦𝑅𝑎 |
14 | 13, 10 | cab 2810 | . . . 4 class {𝑦 ∣ 𝑦𝑅𝑎} |
15 | 9, 14 | cima 5344 | . . 3 class (^{◡}𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) |
16 | 3, 4, 7, 8, 15 | cmpt2 6906 | . 2 class (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^{◡}𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) |
17 | 2, 16 | wceq 1658 | 1 wff ∘_{RV/𝑐}𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^{◡}𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) |
Colors of variables: wff setvar class |
This definition is referenced by: orvcval 31064 |
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