$$ \text{Chapter 8: Continuous random variables} $$

Cumulative distribution function (CDF):

• Only used for continuous variables
• $F(x)=P(X<x)=P(X\le x)$; represents the cumulative probability
• The function is continuously increasing $\implies P(a<X<b)=F(b)-F(a)$

<aside> ⭐ $\displaystyle F(x)=P(X\le {\color{tomato}x})=\int^x_{limit_{\color{TOMATO}l}} f(x)$

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• $F(x)=0$ for $x<limit_l$

• $F(x)=1$ for $x>limit_u$

• $P(X>x)=1-F(x)$

• $P(X<a)=b\implies F(a)=b$

  • Consider carefully which domain $a$ lies in and use that domain to find $a$

• $\displaystyle F(a)=\frac{n}{100}$ where $a$ is the $n^{th}$ percentile

• Convert from PDF to CDF:

• Convert from CDF to PDF:

<aside> ⭐ $\displaystyle f(x)=\frac{dF(x)}{dx}$

</aside>

Mode:

• List all the potential maxima and maxima is $x$ with highest $f(x)$

  • Either a stationary point of 1 domain

    • Find $\displaystyle \frac{df(x)}{dx}=0$ and use $d^2y/dx^2<0$ to prove maximum point

    Or

  • The end point(s) of the domain

    • Sketch graph of all domains to see the highest point

Mean $E(g(X)$:

• Sum all domains after find $E(X)$ for each domain

<aside> ⭐ $\displaystyle E(g(X))=\int_{\forall x}g(x).f(x)dx$ where $g(x)$ is function for change of $f(x)$

</aside>