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Given a uniform distribution on $[0,1]$, the associated density $f(x)=\mathbb 1_{[0,1]}$ and cdf $F(x)=x$ for $x \in [0,1]$ and $=0$ or $=1$ respectively for $x<0$ or $x>1$.

Often I find sources where it says the inverse of the cdf is $F^{-1}(p)=p$. However, for example for $p=1$ it is not unambiguous, i.e. "$F^{-1}(1)=[1, \infty)$".

Is there a convention on how to interpret the inverse of the cdf when reading that the inverse of a cdf is given by such?

Tesla
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1 Answers1

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Here they are using $F^{-1}$ to denote the generalized inverse CDF defined by $F^{-1}(x)=\inf \{t: F(t) \geq x\}$ for $0 \leq x \leq 1$. (Actual inverse function does not exist since $F$ is not bijective on $\mathbb R$). According to this definition you get $F^{-1}(x)=x$ for $0 \leq x \leq 1$.

Kavi Rama Murthy
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