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docs: update education/statistics/random-variable

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education/statistics/random-variable.md
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title: 隨機變數 (Random Variable) title: 隨機變數 (Random Variable)
description: description:
published: true published: true
date: 2025-12-28T17:19:02.040Z date: 2025-12-28T17:19:30.971Z
tags: tags:
editor: markdown editor: markdown
dateCreated: 2025-12-28T16:41:51.938Z dateCreated: 2025-12-28T16:41:51.938Z
@@ -14,50 +14,46 @@ dateCreated: 2025-12-28T16:41:51.938Z
A set of probability value **p~i~** assigned to each of the values taken by the discrete random variables **x~i~**. A set of probability value **p~i~** assigned to each of the values taken by the discrete random variables **x~i~**.
$0 \leq p_i \leq 1$ and $\sum_{p_i} = 1$ and
Probability Probability
$P(X = x) = p_i$
Expectation Expectation
$E(X) = \sum_{i} p_i x_i$
### Cumulative Distribution Function (CDF) ### Cumulative Distribution Function (CDF)
$F(X) = P(X \leq x)$
$F(X) = \sum_{y : y \leq x} P(X = y)$
## Continuous ## Continuous
### Probability Density Function (PDF) ### Probability Density Function (PDF)
Probabilistic properties of a continuous random variable. Probabilistic properties of a continuous random variable.
$\int f(x) \,dx\ = 1$
Expectation Expectation
$E(X) = \int xf(x) \,dx$
### Cumulative Distribution Function (CDF) ### Cumulative Distribution Function (CDF)
$F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy$
$f(X) = \frac{dF(x)}{dx}$
$P(a < x \leq b) = P(X \leq b) - P(X \leq a) = F(b) - F(a)$
$P(a < x \leq b) = P(a \leq x \leq b)$
### Symmetric ### Symmetric
If there is a point that, If there is a point that,
$f(μ + x) = f(μ - x)$
Then, Then,
$E(X) = μ$
Is the expectation of this random variable, equal to the point of symmetry. Is the expectation of this random variable, equal to the point of symmetry.
### Median and Quantiles ### Median and Quantiles
The middle value of the random variable. The middle value of the random variable.
For median, set p to 0.5. For median, set p to 0.5.
$F(X) = p$
### Variance ### Variance
A positive quantity that measures the spread of the distribution of the random variable about its mean value. A positive quantity that measures the spread of the distribution of the random variable about its mean value.
Larger values of the variance indicate that the distribution is more spread out. Larger values of the variance indicate that the distribution is more spread out.
$Var(X) = E(X^{2}) - (E(X))^{2}$ Missing superscript or subscript argument