This gave me a reason (excuse?) to tackle something on my to-do list: learning to use Shiny to create an interactive document containing statistical analysis (or at least statistical mumbo-jumbo). I won't repeat the full discussion here, but instead will link the Shiny document I created. It lets you tweak settings for an example of a reciprocal normal variable and judge for yourself how well various normal approximations fit. I'll just make a few short observations here:
- No way does $Y$ actually have a normal distribution.
- Dividing by $X$ suggests that you probably should be using a distribution with finite tails (e.g., a truncated normal distribution) for $X$. In particular, the original question had $X$ being speed of something, $k$ being (fixed) distance to travel and $Y$ being travel time. Unless the driver is fond of randomly jamming the gear shift into reverse, chances are $X$ should be nonnegative; and unless this vehicle wants to break all laws of physics, $X$ probably should have a finite upper bound (check local posted speed limits for suggestions). That said, I yield to the tendency of academics to prefer tractible/well-known approximations (e.g., normal) over realistic ones.
- The coefficient of variation of $X$ will be a key factor in determining whether approximating the distribution of $Y$ with a normal distribution is "good enough for government work". The smaller the coefficient of variation, the less likely it is that $X$ wanders near zero, where bad things happen. In particular, the less likely it is that $X$ gets anywhere near zero, the less skewness $Y$ suffers.
- There is no one obvious way to pick parameters (mean and standard deviation) for a normal approximation to $Y$. I've suggested a few in the Shiny application, and you can try them to see their effect.