如何在同一直方图中绘制多个函数的密度分布？

核心要点是必须将直方图纵轴设置为密度尺度，否则密度曲线和直方图尺度不匹配无法对齐，你可以根据使用的绘图系统选择对应实现方式。

基础R原生绘图实现 #

• 绘制直方图时指定freq = FALSE，将纵轴从计数转为密度
• 多组直方图叠加时设置填充色半透明，避免完全遮挡下层内容
• 用curve()绘制理论分布密度、lines(density())绘制样本核密度曲线，通过add = TRUE参数叠加到现有画布
• 最后添加图例区分不同分组的图形元素

# 固定随机种子，结果可复现
set.seed(123)
# 生成两组示例分布数据
data_norm1 <- rnorm(1000, mean = 0, sd = 1)
data_norm2 <- rnorm(1000, mean = 2, sd = 1.5)

# 绘制第一组直方图
hist(data_norm1, 
     freq = FALSE, 
     col = rgb(1, 0, 0, 0.25), # 半透红色填充
     xlim = c(-4, 7), ylim = c(0, 0.5),
     main = "直方图叠加多密度曲线示例",
     xlab = "观测值", ylab = "密度")
# 叠加第二组半透直方图
hist(data_norm2, 
     freq = FALSE, 
     col = rgb(0, 0, 1, 0.25), # 半透蓝色填充
     add = TRUE)

# 叠加理论密度曲线
curve(dnorm(x, mean = 0, sd = 1), col = "red", lwd = 2, add = TRUE)
curve(dnorm(x, mean = 2, sd = 1.5), col = "blue", lwd = 2, add = TRUE)
# 叠加样本核密度估计曲线
lines(density(data_norm1), col = "darkred", lwd = 2, lty = 2)
lines(density(data_norm2), col = "darkblue", lwd = 2, lty = 2)

# 添加图例
legend("topright",
       legend = c("N(0,1)样本分布", "N(2,1.5)样本分布", 
                  "N(0,1)理论密度", "N(2,1.5)理论密度",
                  "N(0,1)核密度", "N(2,1.5)核密度"),
       fill = c(rgb(1,0,0,0.25), rgb(0,0,1,0.25), NA, NA, NA, NA),
       col = c(NA, NA, "red", "blue", "darkred", "darkblue"),
       lwd = c(NA, NA, 2, 2, 2, 2),
       lty = c(NA, NA, 1, 1, 2, 2),
       bty = "n")

ggplot2 绘图实现 #

ggplot2语法更简洁，自动生成分组图例，适合快速出出版级图片：

• 先将数据转换为长格式数据框，用分组列映射颜色、填充属性
• 直方图图层指定y = after_stat(density)匹配密度尺度，位置参数设为position = "identity"取消默认堆叠
• 用geom_density()叠加核密度曲线，stat_function()叠加自定义理论密度函数

library(ggplot2)

# 构造长格式示例数据
df <- data.frame(
  value = c(data_norm1, data_norm2),
  group = rep(c("N(0,1)样本", "N(2,1.5)样本"), each = 1000)
)

ggplot(df, aes(x = value, fill = group, color = group)) +
  geom_histogram(aes(y = after_stat(density)), 
                 alpha = 0.3, position = "identity", binwidth = 0.3) +
  geom_density(linewidth = 1, alpha = 0.2, show.legend = FALSE) +
  # 叠加理论密度曲线
  stat_function(fun = dnorm, args = list(mean = 0, sd = 1), 
                color = "darkred", linewidth = 1, linetype = 2) +
  stat_function(fun = dnorm, args = list(mean = 2, sd = 1.5), 
                color = "darkblue", linewidth = 1, linetype = 2) +
  scale_fill_manual(values = c("N(0,1)样本" = "red", "N(2,1.5)样本" = "blue")) +
  scale_color_manual(values = c("N(0,1)样本" = "red", "N(2,1.5)样本" = "blue")) +
  labs(title = "ggplot2版本多密度叠加直方图", x = "观测值", y = "密度") +
  theme_bw()

注意：如果需要叠加其他分布（比如指数分布、伽马分布、自定义分布），只需要替换curve()或stat_function()里的函数名和对应参数即可。

参考效果如下：

内容的提问来源于stack exchange，提问作者Fabricio Quinto