64 lines
1.4 KiB
Markdown
64 lines
1.4 KiB
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: 2025-12-28T17:19:02.040Z
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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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Expectation
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$E(X) = \sum_{i} p_i x_i$
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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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Expectation
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$E(X) = \int xf(x) \,dx$
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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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### Symmetric
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If there is a point that,
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$f(μ + x) = f(μ - x)$
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Then,
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$E(X) = μ$
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Is the expectation of this random variable, equal to the point of symmetry.
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### Median and Quantiles
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The middle value of the random variable.
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For median, set p to 0.5.
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$F(X) = p$
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### Variance
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A positive quantity that measures the spread of the distribution of the random variable about its mean value.
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Larger values of the variance indicate that the distribution is more spread out.
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$Var(X) = E(X^{2}) - (E(X))^{2}$
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