40 lines
872 B
Markdown
40 lines
872 B
Markdown
---
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title: 隨機變數 (Random Variable)
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description:
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published: true
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date: 2025-12-28T17:12:39.067Z
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tags:
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editor: markdown
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dateCreated: 2025-12-28T16:41:51.938Z
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---
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## Discrete
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### Probability Mass Function (PMF)
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A set of probability value **p~i~** assigned to each of the values taken by the discrete random variables **x~i~**.
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$0 \leq p_i \leq 1$ and $\sum_{p_i} = 1$
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Probability
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$P(X = x) = p_i$
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### Cumulative Distribution Function (CDF)
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$F(X) = P(X \leq x)$
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$F(X) = \sum_{y : y \leq x} P(X = y)$
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## Continuous
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### Probability Density Function (PDF)
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Probabilistic properties of a continuous random variable.
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$\int f(x) \,dx\ = 1$
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### Cumulative Distribution Function (CDF)
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$F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy\\$
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$f(X) = \frac{dF(x)}{dx}$
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$P(a < x \leq b) = P(X \leq b) - P(X \leq a) = F(b) - F(a)$
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$P(a < x \leq b) = P(a \leq x \leq b)$
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