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Then, there is an interval [y 1, y 2] containing b, and a region R containing (a, b), such that for every x in R there is exactly one value of y in [y 1, y 2] satisfying ϕ(x, y) = 0, and y is a continuous function of x so that ϕ(x, y(x)) = 0. The total differentials of the functions are:
The independent variable x does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely y(0) = 4, y(−2.7) = 4, y(π) = 4, and so on. No matter what value of x is input, the output is 4. [1] The graph of the constant function y = c is a horizontal line in the plane that passes through the ...
A random variable is said to be truncated from below if, for some threshold value , the exact value of is known for all cases >, but unknown for all cases . Similarly, truncation from above means the exact value of y {\displaystyle y} is known in cases where y < c {\displaystyle y<c} , but unknown when y ≥ c {\displaystyle y\geq c} .
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
z is a free variable and x and y are bound variables, associated with logical quantifiers; consequently the logical value of this expression depends on the value of z, but there is nothing called x or y on which it could depend. More widely, in most proofs, bound variables are used.
In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened ...
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1i + b 2 x 2i + ... + ε i, where y i is the i th observation of the response variable, x ji is the i th observation of the j th ...
A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function. If all vertical lines intersect a curve at most once then the curve represents a function.