A graph plotting the rankits on the horizontal axis and the data points on the vertical axis is called a rankit plot, or, if the distribution is normal, a normal probability plot. Such a plot is necessarily nondecreasing. In large samples from a normally distributed population, such a plot will approximate a straight line. Substantial deviations from straightness are considered evidence against normality of the distribution. Rankit plots are usually used to visually demonstrate whether data are from ...

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Rankit, Rankit - Example, Rankit - Rankit plot, Rankit - Relationship between rankit plots and Q-Q plots, Rankit - History

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Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbers and the central limit theorem. There are various senses in which a sequence (Xn) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables. ...

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Random variable, Random variable - Definitions, Random variable - Random variables, Random variable - Distribution functions, Random variable - Functions of random variables, Random variable - Example, Random variable - Moments, Random variable - Equivalence of random variables, Random variable - Equality in distribution, Random variable - Equality in mean, Random variable - Almost sure equality, Random variable - Equality, Random variable - Convergence, Random variable - Literature

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Here is one of the most familiar realistic examples. Suppose X1, ..., Xn are an independent sample from a normally distributed population with mean μ and variance σ2. Let Then has a Student's t-distribution with n − 1 degrees of freedom. Note that what distribution T has does not depend on the values of the unobservable parameters μ and σ2; i.e., it is a pivotal quantity. If c is t ...

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Confidence interval, Confidence interval - Confidence intervals in measurement, Confidence interval - Robust confidence intervals, Confidence interval - How to understand confidence intervals, Confidence interval - Concrete practical example, Confidence interval - Confidence intervals for proportions and related quantities

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As long as the same conventions are followed at the transition points, the probability density function may also be expressed in terms of the Heaviside step function: and in terms of the rectangle function There is no ambiguity at the transition point of the sign function. Using the half-maximum convention at the transition points, the uniform d ...

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Uniform distribution continuous, Uniform distribution continuous - The cumulative distribution function, Uniform distribution continuous - The moment-generating function, Uniform distribution continuous - Standard uniform, Uniform distribution continuous - Related distributions, Uniform distribution continuous - Relationship to other functions, Uniform distribution continuous - Sampling from a uniform distribution, Uniform distribution continuous - Uses of the uniform distribution

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In the process of weighing 1000 objects, under practical conditions, it is easy to believe that the operator might make a mistake in procedure and so report an incorrect mass (thereby making one type of systematic error). Suppose he has 100 objects and he weighed them all, one at a time, and repeated the whole process ten times. Then he can calculate a sample standard deviation for each object, and look for outliers. Any object with an unusually large standard deviation probably has an outlier in its data. These can be removed by various non ...

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Confidence interval, Confidence interval - Confidence intervals in measurement, Confidence interval - Robust confidence intervals, Confidence interval - How to understand confidence intervals, Confidence interval - Concrete practical example, Confidence interval - Confidence intervals for proportions and related quantities

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Restricting a = 0 and b = 1, the resulting distribution is called a standard uniform distribution. One interesting property of the standard uniform distribution is that if U1 is uniformly distributed, then so is 1-U1: ...

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Uniform distribution continuous, Uniform distribution continuous - The cumulative distribution function, Uniform distribution continuous - The moment-generating function, Uniform distribution continuous - Standard uniform, Uniform distribution continuous - Related distributions, Uniform distribution continuous - Relationship to other functions, Uniform distribution continuous - Sampling from a uniform distribution, Uniform distribution continuous - Uses of the uniform distribution

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An investor might choose to invest a proportion of his wealth in a portfolio of risky assets with the remainder in cash - earning interest at the risk free rate (or indeed may borrow money to fund his purchase of risky assets in which case there is a negative cash weighting). Here, the ratio of risky assets to risk free asset determines overall return - this relationship is clearly linear. It is thus possible to achieve a particular return in one of two ways: By investing all of one’s wealth in a risky portfolio, or by investing a proportion in a risky portf ...

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Capital asset pricing model, Capital asset pricing model - The formula, Capital asset pricing model - Asset pricing, Capital asset pricing model - Asset-specific required return, Capital asset pricing model - Risk and diversification, Capital asset pricing model - The efficient Markowitz frontier, Capital asset pricing model - The market portfolio, Capital asset pricing model - Assumptions of CAPM, Capital asset pricing model - Shortcomings of CAPM, Capital asset pricing model - Finding related topics

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Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function . Proportional to implies that one must multiply or divide by a normalizing constant in order to assign measure 1 to the whole space, i.e., to get a probability measure. In a simple discrete case we have where P(H0) is the prior probability that the hypothesis is true; P(D|H0) is the conditional probability of the data given that the hyp ...

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Normalizing constant, Normalizing constant - Definition and examples, Normalizing constant - Bayes' theorem, Normalizing constant - Non-probabilistic uses

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If X is a discrete random variable taking values on some subset of the non-negative integers, {0,1, ...}, then the probability-generating function of X is defined as: where f is the probability mass function of X. Note that the equivalent notation GX is sometimes used to distinguish between the probability-generating functions of several random variables. ...

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Probability-generating function, Probability-generating function - Definition, Probability-generating function - Properties, Probability-generating function - Power series, Probability-generating function - Probabilities and expectations, Probability-generating function - Functions of independent random variables, Probability-generating function - Examples, Probability-generating function - Related concepts

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The above lead to the following formula for the price of a call on a stock currently trading at price S, where the option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is r and the constant stock volatility is σ. where Here N is the cumulative standard normal distribution function. The price of a put option may be computed from this by put-call ...

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Black-Scholes, Black-Scholes - The model, Black-Scholes - Black-Scholes in practice, Black-Scholes - The formula, Black-Scholes - Extensions of the formula, Black-Scholes - Formula derivation, Black-Scholes - Elementary derivation, Black-Scholes - The Black-Scholes PDE, Black-Scholes - From the general Black-Scholes PDE to a specific valuation, Black-Scholes - Other derivations

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Simulations using the random number generators are often called the Monte Carlo experiments. A common type of this type of an experiment is the generation of random variables with expected mean of 0 and expected variance of 1 and to observe the differences between the expected statistics and the obtained statistics. Results of one of these simulation experiments (with 100,000 generated random variables) are shown below for n equal to 100, 30, 10, 5, and 3. As you may observe, for ns greater than 30, the differences between true and unbiased ...

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True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

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While in practice more advanced models are often used, many of the key insights provided by the Black-Scholes formula have become an integral part of market conventions. For instance, it is common practice for the implied volatility rather than the price of an instrument to be quoted. (All the parameters in the model other than the volatility - that is the time to maturity, the strike, the risk-free rate, and the current underlying price - are unequivocally observable. This means there is one-to-one relationship between the option pri ...

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Black-Scholes, Black-Scholes - The model, Black-Scholes - Black-Scholes in practice, Black-Scholes - The formula, Black-Scholes - Extensions of the formula, Black-Scholes - Formula derivation, Black-Scholes - Elementary derivation, Black-Scholes - The Black-Scholes PDE, Black-Scholes - From the general Black-Scholes PDE to a specific valuation, Black-Scholes - Other derivations

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The above option pricing formula is used for pricing European put and call options on non-dividend paying stocks. The Black-Scholes model may be easily extended to options on instruments paying dividends. For options on indexes (such as the FTSE) where each of 100 constituent companies may pay a dividend twice a year and so there is a payment nearly every business day, it is reasonable to simplify and make the assumption that the dividends are paid continuously. The dividend payment paid over the time period [t,t + dSee also:

Black-Scholes, Black-Scholes - The model, Black-Scholes - Black-Scholes in practice, Black-Scholes - The formula, Black-Scholes - Extensions of the formula, Black-Scholes - Formula derivation, Black-Scholes - Elementary derivation, Black-Scholes - The Black-Scholes PDE, Black-Scholes - From the general Black-Scholes PDE to a specific valuation, Black-Scholes - Other derivations

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The n-1 term in the denominator of the unbiased variance formula is referred to as degrees of freedom, signified as df or by the Greek letter ν. The notion of the degrees of freedom is related to the concept of the random normal variable. To illustrate this concept, let us consider the numbers 0, 1, 2, 3 assigned to five subjects in our illustrative example. These subjects are fictitious, as are the numbers 0, 1, 2, and 3. Don't be misled by their ordinality, as in a recent lot ...

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True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

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The variance can be easily changed from the true variance to the unbiased variance, as and from the unbiased variance to the true variance, as For the example, the true variance (1.25) can be changed to the unbiased variance as (4/3)(1.25) = 1.67 and the unbiased variance (1.67) can be changed to the true variance as (3/4)(1.67) = 1.25. ...

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True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

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Survival models can be usefully viewed as ordinary regression models in which the response variable is time. However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by missing data problems which are peculiar to time. The birth and death of a subject may be known, in which case the lifetime is known. More generally, it may be known only that the date of birth was prior to some date: this is called left censoring. Also, it may be known only that the date of death is ...

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Survival analysis, Survival analysis - General formulation, Survival analysis - Survival function, Survival analysis - Lifetime distribution function and event density, Survival analysis - Hazard function and cumulative hazard function, Survival analysis - Quantities derived from the survival distribution, Survival analysis - Some survival distributions, Survival analysis - Fitting parameters to data

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Survival models are constructed by choosing a basic survival distribution. It is straightforward to phrase model fitting and analysis in general terms, using the concepts outlined in under "General formulation", above. Thus it is relatively easy to substitute one distribution for another, in order to study the consequences of different choices. The choice of survival distribution expresses some particular information about the relation of time and any exogenous variables to survival, and as such, it is analogous to the choice of link ...

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Survival analysis, Survival analysis - General formulation, Survival analysis - Survival function, Survival analysis - Lifetime distribution function and event density, Survival analysis - Hazard function and cumulative hazard function, Survival analysis - Quantities derived from the survival distribution, Survival analysis - Some survival distributions, Survival analysis - Fitting parameters to data

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The existence of life without reproduction is the subject of some speculation. The biological study of how the origin of life led from non-reproducing elements to reproducing organisms is called abiogenesis. Whether or not there were several independent abiogenetic events, biologists believe that the last common ancestor to all present life on earth lived about 3.5 billion years ago. Today, some scientists have speculated about the possibility of creating life non-reproductively in the laboratory. One group of scientists has succeeded ...

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Biological reproduction, Biological reproduction - Asexual reproduction, Biological reproduction - Sexual reproduction, Biological reproduction - Mitosis and Meiosis, Biological reproduction - Reproductive strategies, Biological reproduction - Asexual vs. sexual reproduction, Biological reproduction - The Red Queen hypothesis, Biological reproduction - Life without reproduction

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Using all possible differences between values of a variable as a foundation of statistical theory was contemplated by Kendall (1943, p. 47) who defined a coefficient, called here u², as For the discontinuous infinite case, the above equation can be written as and for the finite case as where the summed term in the above equation is a vector of all possible differences between elements of v ...

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True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

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Given x and y independently uniformly distributed in [−1,1], set R = x2 + y2. If R = 0 or R > 1, throw them away and try another pair (x, y). Then, for these filtered points, compute: and ...

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Box-Muller transform, Box-Muller transform - Basic form, Box-Muller transform - Polar form, Box-Muller transform - Contrasting the two forms

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The CAPM returns the asset-appropriate required return or discount rate - i.e. the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. Betas exceeding one signify more than average "riskiness"; betas below one indicate lower than average. Thus a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. The CAPM is consistent with intuitio ...

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Capital asset pricing model, Capital asset pricing model - The formula, Capital asset pricing model - Asset pricing, Capital asset pricing model - Asset-specific required return, Capital asset pricing model - Risk and diversification, Capital asset pricing model - The efficient Markowitz frontier, Capital asset pricing model - The market portfolio, Capital asset pricing model - Assumptions of CAPM, Capital asset pricing model - Shortcomings of CAPM, Capital asset pricing model - Finding related topics

Read more here: » Capital asset pricing model: Encyclopedia II - Capital asset pricing model - Asset-specific required return

The above definition of variance in terms of differences contained by the data does not involve the arithmetic mean. It seems plausible to assume that the information contained in the above matrix could have been also obtained from a matrix of all possible differences between the data elements and their mean, which can be obtained as Squaring the elements of the above matrix results in a matrix with n columns of squared deviation scores x with column sums (5.00) divided by n (4) equal to the variance computed by divid ...

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True variance, True variance - Computation of the true and unbiased variance, True variance - Changing true variance to unbiased variance and vice versa, True variance - Degrees of freedom, True variance - Degrees of freedom: Monte Carlo simulation, True variance - True variance and all possible differences between values of a variable, True variance - Matrices of differences, True variance - Differences between data elements and their mean, True variance - Variance and Information, True variance - Retrospect, True variance - Conventional language of computation

Read more here: » True variance: Encyclopedia II - True variance - Differences between data elements and their mean