Provide an Appropriate Response

question 98

Essay

Provide an appropriate response.
- $\text { The Cauchy density function, } f ( x ) = \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } \text {, occurs in probability theory. Show that } \int _ { - \infty } ^ { \infty } \frac { 1 } { \pi \left( 1 + x ^ { 2 } \right) } d x = 1 \text {. }$

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Definitions:

Prototype Theory

A theory in cognitive science that suggests concepts or category judgments are made by comparing new stimuli to the "best example" or prototype of that category.

Family Resemblance

The physical or behavioral similarities observed among members of the same family, attributing to shared genetics and environment.

Semantic Features

The components of meaning that come together to define a particular word within linguistic semantics.

Prototype

A typical or optimal example of a concept or category, used as a reference point for comparison, categorization, or understanding.