ece108: add distributions

This commit is contained in:
eggy 2023-03-23 18:53:12 -04:00
parent 701cdb688b
commit 27ca9fa5a4

View File

@ -884,3 +884,30 @@ The **inclusion-exclusion principle** also applies.
$$Pr\{A\cup B\}=Pr\{A\}+Pr\{B\}-Pr\{A\cup B\}$$
### Named PDFs
!!! definition
- An **emperical PDF** is collected from empirical data.
A **Bernouilli trial** is an event with exactly two options, pass $P$ with probability $p$, or fail $F$ with probability $q=1-p$. For the event $X$:
$$
Pr\{X\}=\begin{cases}
p &\text{if }X=\{P\} \\
1-p&\text{if }X=\{F\}
\end{cases}
$$
For exactly two options for $x$ (1 or 0):
$$Pr\{X=x\}=p^x(1-p)^{1-x}$$
Please see [SL Math - Analysis and Approaches 2#Binomial distribution](/g11/mcv4u7/#binomial-distribution) for more information.
A **random variable** is a function that assigns a real number to every item in the sample space. A **discrete random variable** is used if the sample space is discrete. The probability of all events that lead to a possible discrete random variable $x\in\mathbb R$, where $X$ is the function to transform those variables:
$$Pr\{X^{-1}(\{x\})\}$$
Thus the **binomial distribution** for $r$ successes of $n$ total tries, if they are independent, is:
$$Pr\{X=r\}{n\choose r}p^rq^{n-r}$$