Distribution Symmetry Property
For a random variable $X$ that is symmetric about 0, its cumulative distribution function (CDF), denoted $G_X(x)$, satisfies the property:
$G_X(-x) = 1 - G_X(x)$ for all $x$.
Inverse Function Derivation
Let $y = G_X^\leftarrow(p)$. By definition of the inverse CDF (or quantile function), this means:
$G_X(y) = p$
Using the symmetry property $G_X(-y) = 1 - G_X(y)$, we can substitute $p$ for $G_X(y)$:
$G_X(-y) = 1 - p$
Again, applying the definition of the inverse CDF, if $G_X(-y) = 1 - p$, then:
$G_X^\leftarrow(1 - p) = -y$
Now, we need to find the value of the sum $G_X^\leftarrow(p) + G_X^\leftarrow(1 - p)$:
$G_X^\leftarrow(p) + G_X^\leftarrow(1 - p) = y + (-y)$
$G_X^\leftarrow(p) + G_X^\leftarrow(1 - p) = y - y = 0$
Therefore, for a random variable symmetric about 0, the value of $G_X^\leftarrow(p) + G_X^\leftarrow(1 - p)$ is 0.
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