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: 2025-12-28T17:19:02.040Z
+date: 2025-12-28T17:19:30.971Z
 tags: 
 editor: markdown
 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~**.
 
-$0 \leq p_i \leq 1$ 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)
 
-$F(X) = P(X \leq x)$
+  
-$F(X) = \sum_{y : y \leq x} P(X = y)$
 
 ## Continuous
 
 ### Probability Density Function (PDF)
+
 Probabilistic properties of a continuous random variable.
 
-$\int f(x) \,dx\ = 1$
+Expectation  
-
-Expectation
-$E(X) = \int xf(x) \,dx$
 
 ### Cumulative Distribution Function (CDF)
 
-$F(X) = P(X \leq x) = \int_{-\infty}^{x} f(y) \,dy$
+  
-$f(X) = \frac{dF(x)}{dx}$
+  
-$P(a < x \leq b) = P(X \leq b) - P(X \leq a) = F(b) - F(a)$
+  
-$P(a < x \leq b) = P(a \leq x \leq b)$
 
 ### Symmetric
-If there is a point that,
+
-$f(μ + x) = f(μ - x)$
+If there is a point that,  
-Then,
+  
-$E(X) = μ$
+Then,  
+  
 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.
+The middle value of the random variable.  
-$F(X) = p$
+For median, set p to 0.5.  
 
 ### 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.
 
-$Var(X) = E(X^{2}) - (E(X))^{2}$
+Missing superscript or subscript argument