Is there data that suggests testosterone would work as a male birth control?
The hard part is balancing the sterilisation effect with other side effects whilst having good efficacy. And then there's also the problem of reversing the effects.
I'm not on the pharmaceutical business, there's probably a good reason it doesn't exist yet. I just don't know if it's a legal/public perception thing regarding anabolic steroids, it's too hard to manage the side effects or if it's something else.
Is sex education that bad? Do people think it's made on demand?
It was a great example for me of mass ignorance. Tons of stuff, including this, are very well established, but since we live is such an individualized society where interactions with kids and parents are no longer something that happens at all ages continuously throughout life, and your similarly aged friends for the same reason neither, could it happen that such facts simply never pop up until somebody runs into them.
In 30 years, you'll not believe established science about aging.
If we take it to be a simple exponential distribution with mean 66 days, then lambda = 1/66. and variance is 4356 or a standard deviation of 66 days as well.
In that case, by month 4, about 83% of men would be expected to have sperm in their semen.
To delve deeper, what you're missing here is the idea of skewness and kurtosis. These are the 'higher moments' of the probability distribution function (natural extensions of mean, variance, etc). In particular, the skewness is where you're getting hung up. You assume in your first sentence that the skewness of every pdf is zero (i.e., it 'goes both ways'). That is not generally the case.
In particular, the skewness of the 'time to first sperm' distribution must be very positive because it can never be below zero.
If you were talking about how much sperm they had exponential would indeed be better. But we’re not. Just if they have any.
If the average time it takes is 2,5 months, you should expect to wait on average 2,5 months, not 4.
That is indeed not the same as asking when do 83% of men have sperm, since 83% is not the average.
Things cannot be distributed where the distribution has values the thing can never have. The time until an event cannot be negative. The cdf of a normal distribution always has a non-zero value at zero. There is no chance a sperm is ejaculated before it's made. That's a logical impossibility. Thus time to first sperm is absolutely not normally distributed.
It would not be tested explicitly for such because it is both bad at its job and has negative side effects well known from all the steroid users.