class: center, middle

Gerard Escudero, 2020

.footnote[source ]

class: left, middle, inverse

• .cyan[Probability]

• Normal Distribution

• References

.blue[How likely something is to happen.]

Values: from .blue[0] (impossible) to .blue[1] (certain)

$$P(\text{event})=\frac{\text{Outcomes event}}{\text{Total outcomes}}$$

Examples:

.cols5050[ .col1[

• .blue[dice]

$$P(1)=\frac{1}{6}$$

• .blue[coin]

$$P(H)=\frac{1}{2}$$ ] .col2[

• .blue[king from a deck]

$$P(king)=\frac{4}{52}$$ ]]

.blue[Complement]:

.cols5050[ .col1[ $$P(A)+P(A')=1$$ ] .col2[ .center[
source ]]]

.blue[Independent]:

$$P(\text{1 or 2})=P(\text{1})+P(\text{2})=\frac{2}{6}$$

$$P(\text{{1,1}})=P(\text{1})*P(\text{1})=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}$$

.cols5050[ .col1[ $$P(\text{2 blues})=\frac{2}{5}\times\frac{1}{4}=\frac{2}{20}$$

$$P(\text{blue})=\frac{2}{5}$$

$$P(\text{blue}|\text{blue})=\frac{1}{4}$$

] .col2[ .center[
source ]]]

.blue[Conditional Probability]: $P(A|B)$ $\rightarrow$ "A given B"

.cols5050[ .col1[ $$P(A|B)=\frac{P(A)P(B|A)}{P(B)}$$ ] .col2[ $P(\text{man})=0.4$
$P(\text{pink})=0.25$
$P(\text{pink}|\text{man})=0.125$ ]]

$$P(man|pink)=\frac{P(man)P(pink|man)}{P(pink)}=\frac{0.4\times 0.125}{0.25}=0.2$$

.blue[Principal component] of probabilistic Machine Learning algorithms.

.footnote[source]

class: left, middle, inverse

• .brown[Probability]

• .cyan[Normal Distribution]

  • Measures

  • Distributions

  • Standardization

• References

.cols5050[ .col1[ .center[ ]] .col2[ Examples:

• heights of people
• size of things produced by machines (sugar)
• ... ]]

.blue[Mean]: average of the numbers

$$\mu=\frac{\sum_{i=1}^n x_i}{n}$$

.blue[Standard Deviation]: a measure of how spread out numbers are

.cols5050[ .col1[ $$\sigma=\sqrt{\frac{\sum_{i=1}^n (x_i-\mu)^2}{n}}$$ ] .col2[ $\pm1\sigma\rightarrow68%$
$\pm2\sigma\rightarrow95%$
$\pm3\sigma\rightarrow99.7%$ ]] .center[] .footnote[source]

.blue[z-score]:

$$z=\frac{x-\mu}{\sigma}$$

.blue[Standardization]: normalization

.center[
source ]

• Standard Normal Distribution Table

• Practical example

• MathIsFun.com. Using and Handling Data. Math is Fun, 2019.
  
  
  

$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$