docs: update education/statistics/random-variable

education/statistics/random-variable.md

@@ -2,7 +2,7 @@
 title: 隨機變數 (Random Variable)
 description: 
 published: true
-date: 2025-12-28T17:12:39.067Z
+date: 2025-12-28T17:19:02.040Z
 tags: 
 editor: markdown
 dateCreated: 2025-12-28T16:41:51.938Z
@@ -19,6 +19,9 @@ $0 \leq p_i \leq 1$ and $\sum_{p_i} = 1$
 Probability
 $P(X = x) = p_i$
 
+Expectation
+$E(X) = \sum_{i} p_i x_i$
+
 ### Cumulative Distribution Function (CDF)
 
 $F(X) = P(X \leq x)$
@@ -31,9 +34,30 @@ Probabilistic properties of a continuous random variable.
 
 $\int f(x) \,dx\ = 1$
 
+Expectation
+$E(X) = \int xf(x) \,dx$
+
 ### Cumulative Distribution Function (CDF)
 
-$F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy\\$
+$F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy$
 $f(X) = \frac{dF(x)}{dx}$
 $P(a < x \leq b) = P(X \leq b) - P(X \leq a) = F(b) - F(a)$
 $P(a < x \leq b) = P(a \leq x \leq b)$
+
+### Symmetric
+If there is a point that,
+$f(μ + x) = f(μ - x)$
+Then,
+$E(X) = μ$
+Is the expectation of this random variable, equal to the point of symmetry.
+
+### Median and Quantiles
+The middle value of the random variable.
+For median, set p to 0.5.
+$F(X) = p$
+
+### Variance
+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.
+
+$Var(X) = E(X^{2}) - (E(X))^{2}$