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today we're going to learn about the asymptotic equi-partition property aep it's a fancy name but relies on a few familiar concepts so we're going to make the overview as clear as possible using cats the asymptotic equi-partition property is an important theorem in information theory it relates to sequences of random events like a sequence of coin flips or a cat which spends each day either mounting or sleeping when you have a really long sequence of the random events there are a lot of possible outcomes but aap says that among those many possible sequences only a small subset will occur in practice although each of those typical sequences is equally likely this theorem is used in compression where it allows us to come up with efficient codes for sending bits across a channel aep is also used in error correction for reliable data communication where we can detect if a faulty device accidentally flipped a bit and recover what the message was intended to be on our channel we already have a video that gives a short overview of aep with a cat named ella in this video we will continue with our friends ashley ella and bill but we delve deeper into the mathematical concepts behind aep and present a proof it'll be helpful if you already have a basic understanding of probability so feel free to pause watch one of the linked videos and come back to ours alright so quick recap of our characters we have two siblings ashley and bill they have a super cute cat named ella now they're stuck at home because there's a global pandemic happening and ella meows all the time although occasionally she has a really sleepy quiet day and doesn't meow at all one day ashley and bill decide to start keeping track of the days when ella meows m or sleeps s call this random variable x i for day i they decide it's reasonable to assume the xi are independently and identically distributed this means whether ella meows or sleeps today is not affected by what you did yesterday or any time in the past whatever tells her deep down inside how to behave doesn't change from day to day they notice that about 10 of the time ella spends the whole day sleeping more often she's mowing away we can write this mathematically as the probability that x i equals m is 0.9 the probability that x i equals s is 0.1 now it's one thing to compute the probability of ella meowing or sleeping on a single day but what about if we want to figure out how likely a sequence of days is this is called computing the joint probability because the x i are independent we can split the probability into a product of probabilities for each random variable the sequence of 10 days in a row where lmos every day has the probability 0.9 to the 10 which is about...