In (f) above, I mentioned that Bitcoin miners around the world are currently per- forming evaluations of the hash functi

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In F Above I Mentioned That Bitcoin Miners Around The World Are Currently Per Forming Evaluations Of The Hash Functi 1 (48.72 KiB) Viewed 85 times

In (f) above, I mentioned that Bitcoin miners around the world are currently per- forming evaluations of the hash function at a total rate of 5/3 x 1018 per second. This means that Bitcoin mining operates in a regime in which each individual hash calculation is performed very quickly, but also has a very low probability of being less than the threshold l. I draw your attention to the fact that this is well-modelled by the limiting regime that we used in a number of our derivations in lectures. (a) Assume that a block has just been mined, and let T be the time in seconds until the next block is mined somewhere in the world. Carefully explaining your reasoning, name the distribution of the random variable T, give the values of its parameter(s) and specify its mean and variance.
(b) Assume that a block was mined three minutes ago, and let U be the time in seconds until the next block is mined somewhere in the world. Again carefully explaining your reasoning, name the distribution of the random variable U, give the values of its parameter(s) and specify its mean and variance.
(c) What property of the random variable T explains the observation that you made in (b)? (d) Let J be the number of blocks that are mined in an hour worldwide. Name the distribution of the random variable J, give the values of its parameter(s) and specify its mean and variance.
(e) Let S be the length of time, in seconds, that the vendor in (h) above has to wait before six blocks are mined worldwide. Name the distribution of the random variable S, give the values of its parameter(s) and specify its mean and variance.
(f) Most Bitcoin miners work for mining pools. Assume that a particular pool, Bitcoin Mining Inc., controls a proportion p of the world's total computing power devoted to Bitcoin mining and let JA be the number of blocks mined by Bitcoin Mining Inc in an hour. Conditional on the random variable J in (d) being equal to j, what is the distribution of JA? Use the Theorem of Total Probability to derive the distribution of JA and hence its mean and variance.