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Movements, leaders, victories and the continuing fight for equality.
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THE BHP WIRE —
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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
Innovation, patents, science, technology and world-changing contributions.
Pioneers, champions, Negro Leagues, records, activism and excellence.
Meet the people whose lives, choices and achievements shaped the journey.
Black towns, communities, institutions and places where history happened.
Moments that changed communities, movements, institutions and the nation.
In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In probability and statistics, the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle {\mathcal {D}}} is the function Q {\displaystyle Q} such that Pr [ X ≤ Q ( p ) ] = p {\displaystyle \Pr \left[\mathrm {X} \leq Q(p)\right]=p} for any random variable X ∼ D {\displaystyle \mathrm {X} \sim {\mathcal {D}}} and probability p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} . The quantile function is also called the percentile function (after the percentile), percent-point function, inverse cumulative distribution function or inverse distribution function.
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups). The groups created are termed halves, thirds, quarters, etc., though more often the terms for the quantile are used for the groups created, rather than for the cut points. q-quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 subsets of the q-quantiles, one for each integer k satisfying 0 < k < q. In some cases the value of a quantile may not be uniquely determined, as can be the case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous distributions, providing a way to generalize rank statistics to continuous variables (see percentile rank). When the cumulative distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of the cumulative distribution function) to the values {1/q, 2/q, …, (q − 1)/q}.
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u {\displaystyle u} between 0 and 1, interpreted as a probability, and then returns the smallest number x ∈ R {\displaystyle x\in \mathbb {R} } such that F ( x ) ≥ u {\displaystyle F(x)\geq u} for the cumulative distribution function F {\displaystyle F} of a random variable. For example, imagine that F {\displaystyle F} is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution. We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs to the left of that number. Intuitively, we are unlikely to choose a number in the far end of tails because there is very little area in them which would require choosing a number very close to zero or one. Computationally, this method involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function (CDF) of the distribution (which maps a number in the domain to a probability between 0 and 1) and then inverting that function. This is the source of the term "inverse" or "inversion" in most of the names for this method. Note that for a discrete distribution, computing the CDF is not in general too difficult: we simply add up the individual probabilities for the various points of the distribution. For a continuous distribution, however, we need to integrate the probability density function (PDF) of the distribution, which is impossible to do analytically for most distributions (including the normal distribution). As a result, this method may be computationally inefficient for many distributions and other methods are preferred; however, it is a useful method for building more generally applicable samplers such as those based on rejection sampling. For the normal distribution, the lack of an analytical expression for the corresponding quantile function means that other methods (e.g. the Box–Muller transform) may be preferred computationally. It is often the case that, even for simple distributions, the inverse transform sampling method can be improved on: see, for example, the ziggurat algorithm and rejection sampling. On the other hand, it is possible to approximate the quantile function of the normal distribution extremely accurately using moderate-degree polynomials, and in fact the method of doing this is fast enough that inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.
In statistics, the logit (logistic unit) or log-odds function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations. Mathematically, the logit is the inverse of the standard logistic function σ ( x ) = 1 / ( 1 + e − x ) {\displaystyle \textstyle \sigma (x)=1/(1+e^{-x})} , so the logit is defined as logit p = σ − 1 ( p ) = ln p 1 − p for p ∈ ( 0 , 1 ) . {\displaystyle \operatorname {logit} p=\sigma ^{-1}(p)=\ln {\frac {p}{1-p}}\quad {\text{for}}\quad p\in (0,1).} Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds p 1 − p {\textstyle {\frac {p}{1-p}}} where p is a probability. Thus, the logit is a type of function that maps probability values from ( 0 , 1 ) {\displaystyle (0,1)} to real numbers in ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} , akin to the probit function.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Brown v. Board of Education in 1954.
