ece108: add distributions
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@ -884,3 +884,30 @@ The **inclusion-exclusion principle** also applies.
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$$Pr\{A\cup B\}=Pr\{A\}+Pr\{B\}-Pr\{A\cup B\}$$
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### Named PDFs
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!!! definition
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- An **emperical PDF** is collected from empirical data.
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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 $X$:
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$$
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Pr\{X\}=\begin{cases}
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p &\text{if }X=\{P\} \\
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1-p&\text{if }X=\{F\}
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\end{cases}
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$$
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For exactly two options for $x$ (1 or 0):
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$$Pr\{X=x\}=p^x(1-p)^{1-x}$$
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Please see [SL Math - Analysis and Approaches 2#Binomial distribution](/g11/mcv4u7/#binomial-distribution) for more information.
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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:
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$$Pr\{X^{-1}(\{x\})\}$$
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Thus the **binomial distribution** for $r$ successes of $n$ total tries, if they are independent, is:
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$$Pr\{X=r\}{n\choose r}p^rq^{n-r}$$
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