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I read a post that says:

"Math definition is that the quantile function is the inverse of the distribution function at α. It specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability:here

(1) $P(X < F^{-1}(\alpha))$ = $\Phi^{-1} (\alpha) = \alpha$

Where F⁻¹(α) denotes the α quantile of X.

I think there is a mistake in equation (1). That is, should they said that:

$\Phi^{-1} (\alpha) = x$

$F(x)= P(X < x) = \alpha$,

$F^{-1}(\alpha)= x$

hence,

$F^{-1}(\alpha) = \Phi^{-1} (\alpha) = P(X < F^{-1}(\alpha))= x$

Maryam
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  • I am not sure of understanding what your question is. Could you be more precise ? – Adrien Jul 31 '21 at 14:44
  • @Adrien I think there is a mistake in the post: "$F^{-1}(\alpha) = P(X< F^{-1}(\alpha)) = x$" The inverse should give `x` not $\alpha$ – Maryam Jul 31 '21 at 14:49

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