随机变量函数的期望值和方差
如果X是离散随机变量，g是函数，那么g(X)也是离散随机变量。本节将学习如何计算随机变量函数的期望值和方差，这是概率论中的重要技能。
If X is a discrete random variable, and g is a function, then g(X) is also a discrete random variable. This section will teach you how to calculate the expected value and variance of functions of random variables, which is an important skill in probability theory.
对于简单的函数，如加法和乘以常数，我们可以使用特定的规则来简化计算。这些规则在实际应用中非常有用，能够大大提高计算效率。
For simple functions, such as addition and multiplication by a constant, we can use specific rules to simplify calculations. These rules are very useful in practical applications and can greatly improve computational efficiency.
对于随机变量X的函数g(X)，其期望值E(g(X)) = Σg(x)P(X = x)。这是E(X²)公式的更一般形式。
For a function g(X) of random variable X, the expected value E(g(X)) = Σg(x)P(X = x). This is a more general version of the formula for E(X²).
如果X是随机变量，a和b是常数，则E(aX + b) = aE(X) + b。这个规则大大简化了线性函数的期望值计算。
If X is a random variable and a and b are constants, then E(aX + b) = aE(X) + b. This rule greatly simplifies the calculation of expected values for linear functions.
如果X是随机变量，a和b是常数，则Var(aX + b) = a²Var(X)。注意常数项不影响方差。
If X is a random variable and a and b are constants, then Var(aX + b) = a²Var(X). Note that constant terms do not affect variance.
\[E(g(X)) = \sum g(x) P(X = x)\]
随机变量X的函数g(X)的期望值。
The expected value of function g(X) of random variable X.
\[E(aX + b) = aE(X) + b\]
其中a和b是常数。
Where a and b are constants.
\[Var(aX + b) = a^2 Var(X)\]
其中a和b是常数，常数项不影响方差。
Where a and b are constants, constant terms do not affect variance.
\[E(X + Y) = E(X) + E(Y)\]
如果X和Y是随机变量。
If X and Y are random variables.