# 90% confidence interval for mining a block?

I did some math and found that the 90% confidence interval for bitcoin blocks is that the block will take less than 1381.6 seconds. I used the calculation ln(10)*600 with 10 being used because a 90% confidence interval means one in ten blocks are expected to fall outside of this range. Could this be considered a 90% confidence interval?

Note: this could change if the difficulty is expected to rise or drop. If block time is 576 seconds, for example, it becomes a 91% confidence interval. In practice this changes a lot once the difficulty adjustment has passed.

• Note that the network difficulty is only retargeted every 2016 blocks, so any changes to the amount of hashpower pointed at the network in between retargeting events will alter the effective confidence interval. Historically, hashrate has increased on average, and thus we may expect the measured result for the 90% confidence interval to be less than predicted, on average. Aug 21, 2020 at 19:59
• I don't think you understand correctly the definition of a confidence interval. It is a range of values where you can be e.g. 90 percent confident the interval contains the population mean. Typically you will calculate the confidence interval using sample data e.g. sample mean, sample size. You have not given any sample data. Instead we know the population mean assuming constant hash rate. It is 10 minutes. Hence I have given quantiles of the exponential distribution below in my answer. Aug 23, 2020 at 9:38

Bitcoin blocks are mined according to a Poisson process with a mean of 10 minutes (600 seconds) if you assume constant network hash rate. The difficulty of mining blocks gets adjusted every 2016 blocks to address this variability.

Interarrival times of a Poisson process are exponentially distributed.

The 95th quantile of an exponential distribution with a parameter of 1/600 is 1797.4

The 5th quantile of an exponential distribution with a parameter of 1/600 is 30.8.

90 percent of the time an observation from an exponential distribution will fall in the interval of (30.8,1797.4) seconds

The median time is 415.9 seconds.

Ln doesn't make sense, you need to use gamma distribution. On https://keisan.casio.com/exec/system/1180573218 with 0.95, 2, 10 I get 47 minutes.

• It said 90 not 95. Aug 21, 2020 at 10:58
• @NumberFile Hence I wrote 0.95. On simplypsychology.org/confidence-interval.html as you can see for 5%, 2.5% is on the right 2.5% is on the left. If he had said 80% interval I would be doing 90% for the upper bound and 10% for the lower bound. Aug 21, 2020 at 11:07