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Similarly, X − takes value 2k with probability 6(2kπ) −2 for each positive integer k and takes value 0 with remaining probability. Using the definition for non-negative random variables, one can show that both E[X +] = ∞ and E[X −] = ∞ (see Harmonic series). Hence, in this case the expectation of X is undefined.
Taking the derivative of P(x) with respect to , setting it to 0, and solving for x, we find that the optimal x is equal to 1/e. Thus, the optimal cutoff tends to n/e as n increases, and the best applicant is selected with probability 1/e. For small values of n, the optimal r can also be obtained by standard dynamic programming methods.
In probability theory, Wald's equation, Wald's identity [1] or Wald's lemma [2] is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities.
The binomial test is useful to test hypotheses about the probability of success: : = where is a user-defined value between 0 and 1.. If in a sample of size there are successes, while we expect , the formula of the binomial distribution gives the probability of finding this value:
The probability generating function of an almost surely constant random variable, i.e. one with (=) = and () = is G ( z ) = z c . {\displaystyle G(z)=z^{c}.} The probability generating function of a binomial random variable , the number of successes in n {\displaystyle n} trials, with probability p {\displaystyle p} of success in each trial, is
The idea behind Chauvenet's criterion finds a probability band that reasonably contains all n samples of a data set, centred on the mean of a normal distribution.By doing this, any data point from the n samples that lies outside this probability band can be considered an outlier, removed from the data set, and a new mean and standard deviation based on the remaining values and new sample size ...
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. [1] The formula is still used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data.
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...