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In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests. It asks the following question: if each box of a given product (e.g., breakfast cereals) contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought ...
Conditional expectation. In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can ...
The violet is the mutual information . In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys. The entropy of conditioned on is written as .
v. t. e. In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable , which takes values in the set and is distributed according to , the entropy is where denotes the sum over the variable's possible ...
This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values. [19] Moreover, if given a random variable with finitely or countably many possible values, the Lebesgue theory of expectation is identical to the summation formulas given above.
Probability theory. In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. [ 1] This particular method relies on event A occurring with some sort of relationship with another event B.
Hyperbolic discounting is mathematically described as. where g ( D) is the discount factor that multiplies the value of the reward, D is the delay in the reward, and k is a parameter governing the degree of discounting (for example, the interest rate ). This is compared with the formula for exponential discounting:
Conditional probability distribution. In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables and , the conditional probability distribution of given is the ...