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World population pyramid from 1950 to projected in 2100 (UN, World Population Prospects 2017) A population pyramid (age structure diagram) or "age-sex pyramid" is a graphical illustration of the distribution of a population (typically that of a country or region of the world) by age groups and sex; it typically takes the shape of a pyramid when the population is growing. [1]
Type I or convex curves are characterized by high age-specific survival probability in early and middle life, followed by a rapid decline in survival in later life. They are typical of species that produce few offspring but care for them well, including humans and many other large mammals such as elephants .
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
Population structure may refer to many aspectsof population ecology: Population structure (genetics) , also called population stratification Population pyramid
Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class. The convexity of a measure μ on n-dimensional Euclidean space R n in the sense above is closely related to the convexity of its probability density function. [2]
If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode. Note that under this definition the uniform distribution is unimodal, [4] as well as any other distribution in which the maximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Usually this definition allows ...
The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact ...