"Density" curve overlay on histogram where vertical axis is frequency (aka count) or relative frequency?

@joran's response/comment got me thinking about what the appropriate scaling factor would be. For posterity's sake, here's the result.

When Vertical Axis is Frequency (aka Count)

Thus, the scaling factor for a vertical axis measured in bin counts is

In this case, with N = 164 and the bin width as 0.1, the aesthetic for y in the smoothed line should be:

y = ..density..*(164 * 0.1)

Thus the following code produces a "density" line scaled for a histogram measured in frequency (aka count).

df1            <- data.frame(v = rnorm(164, mean = 9, sd = 1.5))
b1             <- seq(4.5, 12, by = 0.1)
hist.1a        <- ggplot(df1, aes(x = v)) + 
                    geom_histogram(aes(y = ..count..), breaks = b1, 
                                   fill = "blue", color = "black") + 
                    geom_density(aes(y = ..density..*(164*0.1)))
hist.1a

When Vertical Axis is Relative Frequency

Using the above, we could write

hist.1b        <- ggplot(df1, aes(x = v)) + 
                    geom_histogram(aes(y = ..count../164), breaks = b1, 
                                   fill = "blue", color = "black") + 
                    geom_density(aes(y = ..density..*(0.1)))
hist.1b

When Vertical Axis is Density

hist.1c        <- ggplot(df1, aes(x = v)) + 
                    geom_histogram(aes(y = ..density..), breaks = b1, 
                                   fill = "blue", color = "black") + 
                    geom_density(aes(y = ..density..))
hist.1c

Try this instead:

ggplot(df1,aes(x = v)) + 
   geom_histogram(aes(y = ..ncount..)) + 
   geom_density(aes(y = ..scaled..))