The Value Of $x ^ { 2 }$ With $v$ Degrees of Freedom, Such That the Area to Its Left

The value of $x ^ { 2 }$ with $v$ degrees of freedom, such that the area to its left under the chi-squared curve is equal to A, is denoted by $\bar { x } _ { A , v } ^ { 2 }$ , while $x _ { 1 - A , v } ^ { 2 }$ denotes the value such that the area to its right is A.