$$\definecolor{c1}{RGB}{255, 0, 0} \definecolor{c2}{RGB}{0, 0, 255} \definecolor{c3}{RGB}{255, 165, 0} \definecolor{c4}{RGB}{75, 0, 130} \definecolor{c5}{RGB}{220, 220, 0} \definecolor{c6}{RGB}{238, 130, 238} \definecolor{c7}{RGB}{0, 128, 0} \definecolor{c8}{RGB}{100,100,100} \definecolor{c9}{RGB}{45,177,93} \definecolor{default}{RGB}{10,20,20} \color{default}$$ # probability_distribution %%visits: 2 ## introduction the probability_distribution function maps a set to the interval [0, 1] ## intuition A Cumulative distribution near −∞ the function is zero. And near ∞ is 1. Which makes sense as it is a summation of probabilities, which is positive.
It is Also strictly increasing, again, since it is a summation of probabilities, which is positive.
It is also right continuous, see figure below, as the value at the discrete point is defined.
{{file:./screenshot_20210930_141201.png}}
A probability_distribution must obey the following laws. - limx → −∞F(x) = 0 - limx → ∞F(x) = 1 - F is monotone increasing; - F is right continuous, limy+ → xF(y) = F(x) Think of the discrete case.
FX(x) = ∫0∞f(x)dx
What is the cumulative distribution function of a positive random variable X that has probability density function f(x) = 4xe−2x, for x > 0. - Integration by parts, ∫0∞4xe−2xdx = −2xe−2x − e−2x
What is the formula for the pdf of a binomial distribution? - X Bin(n, p) $\mathbb{P}(X=x) = \begin{pmatrix} n \\ x \end{pmatrix} p^{x} (1-q)^{n-x}$ - Where $nCr = \frac{n!}{r!\left( n-r \right) !}$ ## probability_density_function A probability density function f is defined by the three properties it has - ℙ(a < f < b) = ∫abf(x)dx - ∀x in the domain, f(x) ≥ 0 - ∫−∞∞ = 1 ## resources tags :math:
X Bin(n, p) $$ \color{c4}\mathbb{P}(X=x) = \color{c1} \begin{pmatrix} n \\ x \end{pmatrix} \color{c2}p^{x} \color{c3}(1-q)^{n-x} $$ $$ \color{c4}\text{The chance that an event occurs exactly x times is } \color{c2} \text{the chance of the event occurring x times }\color{c3} \text{with the chance of the event not occurring any other time }\color{c1} \text{while considering all combinations where the events could occur. }\color{default} $$
marginal distributio := ∑x1, x2, x3, x4…xnℙ(x1 = y1, …, xk − 1 = yk − 1, xk = yk)
:~ Summing over all possible values of the remaining r.v.s With the
joint distribution we can compute the marginals, but with the marginals,
we can’t say much (unless independent). When we compute independence, we
assume the existence of a mulitvariate
[[../probability_distribution|probability_distribution]]
What is covariance? cov(X, Y) = 𝔼(XY) − 𝔼(X)𝔼(Y)
cov(X, Y) = 𝔼[(X − 𝔼(X))(Y − 𝔼(Y))]
$f_{Y}(y) = f_{X}(g^{-1}(y))\frac{d}{d y} g^{-1}$
$corr(X,Y) = \frac{cov(X,Y)}{\sqrt{Var(X)Var(Y)} }$
Cauchy-Schwarz Inequality ∀X, Y|Corr(X, Y)| ≤ 1
$\mathbb{E}(h(X)|Y=y) = \inth(x) f_{X|Y}(x|y) dx$
$f_{X|Y}(x|y) = \frac{f_{X|Y}(x|y)}{f_{Y}(y)}$
Independence ⟹ Covariance = 0
$f(x) = \frac{1}{\pi}\frac{1}{1+(x-\theta)} ; -\infty<x<\infty$
$\int_{-\infty}^{\infty} \frac{1}{1+y^2}$
$\int \frac{2y}{1+y^2}$
t = 1 + y2
$\frac{dt}{dy} 2y$
$= \frac{dt}{t} = \log (t)$
tags: :probability_and_statistics_2: