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- Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently. (en)
- Suppose a new dice factory has just been built. The first few dice come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired uniform distribution. (en)
- Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be quite certain that one day the number will become zero, and will stay zero forever after. (en)
- Let be the fraction of heads after tossing up an unbiased coin times. Then has the Bernoulli distribution with expected value and variance . The subsequent random variables will all be distributed binomially. (en)
- We may be almost sure that one day this amount will be zero, and stay zero forever after that. (en)
- Suppose that a random number generator generates a pseudorandom floating point number between 0 and 1. Let random variable represent the distribution of possible outputs by the algorithm. Because the pseudorandom number is generated deterministically, its next value is not truly random. Suppose that as you observe a sequence of randomly generated numbers, you can deduce a pattern and make increasingly accurate predictions as to what the next randomly generated number will be. Let be your guess of the value of the next random number after observing the first random numbers. As you learn the pattern and your guesses become more accurate, not only will the distribution of converge to the distribution of , but the outcomes of will converge to the outcomes of . (en)
- Consider the following experiment. First, pick a random person in the street. Let be their height, which is ex ante a random variable. Then ask other people to estimate this height by eye. Let be the average of the first responses. Then by the law of large numbers, the sequence will converge in probability to the random variable . (en)
- Suppose is an iid sequence of uniform random variables. Let be their sums. Then according to the central limit theorem, the distribution of approaches the normal distribution. This convergence is shown in the picture: as grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.
center|200px (en)
- Let X1, X2, … be the daily amounts the charity received from him. (en)
- As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely. (en)
- However, when we consider any finite number of days, there is a nonzero probability the terminating condition will not occur. (en)
- As grows larger, this distribution will gradually start to take shape more and more similar to the bell curve of the normal distribution. If we shift and rescale appropriately, then will be converging in distribution to the standard normal, the result that follows from the celebrated central limit theorem. (en)
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