Axioms:

• For each subset $A$ such that $A \subset \Omega$. $P(A)\ge 0$
• For all disjoint subsets $A$ and $B$, the probability of $A$ or $B$ is: $P(A\cup B)=P(A)+P(B)$
• $P(\Omega)=1$

Conditional probability

$$ P(A \mid B) = \frac{P(A \cap B)}{P(B)} $$

• Bayes’ theorem

$$ P(A \mid B) = \frac{I(A) \, \mathcal{L}(B \mid A)}{E(B)} $$

• $I(A)$ the prior probability of $A$
• $\mathcal{L}(B|A)$ the likelihood probability of $B$ conditional to $A$
• $E(B)$ the evidence of $B$

$$ \begin{aligned} E(B) &= \sum_i I(A_i) \mathcal{L}(B \mid A_i) \\ &= \int I(A) \mathcal{L}(B \mid A) \, {\rm d}A \end{aligned} $$

Probability distributions

• Continuous probability distribution

$$ \int_\Omega\overbrace{f(x)}^{\text{PDF}}\,\text dx=1 $$

• Discrete probability distribution

$$ \sum_{X\in\Omega}f_X(x)=1 $$

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/bucket_1faa3.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/bucket_1faa3.png" width="40px" /> Binomial distribution:

$$ P(k, n, p) = p^k (1-p)^{(n-k)} \frac{n!}{k!(n-k)!} \, $$

• $k$ is the number of successes
• $n$ is the number of trials/events
• $p$ is the probability of an individual success

Properties:

• Mean: $np$
• Variance: $np(1-p)$ </aside>

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/tropical-fish_1f420.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/tropical-fish_1f420.png" width="40px" /> Poisson distribution:

$$ P(k, \lambda) = e^{-\lambda} \frac{\lambda^k}{k!} $$

• $k$ the number of times an event occurs in a given interval
• $\lambda$ the average event rate per interval ($\lambda\in|\R|)$

Properties:

• Mean: $\lambda$
• Variance: $\lambda$ </aside>

<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/coin_1fa99.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/coin_1fa99.png" width="40px" /> Gaussian distribution:

$$ P(x, \mu, \sigma) \sim \mathcal{N}(\mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \,, $$

• $\mu$ is the mean of the Gaussian
• $\sigma$ is the standard distribution of the gaussian

Properties:

• Mean, median, mode: $\mu$
• Variance $\sigma^2$

📝 Note: $\sigma$ areas for the distribution are $68.26,95.44,99.72$

</aside>

Gaussian distribution limit