# Proof that Quantile Function characterizes Probability Distribution

The quantile function is defined as $Q(u)= \inf \{x: F(x) \geq u\}$.

It is well known the distribution function characterizes the probability distribution in the following sense

Theorem Let $X_{1}$ and $X_{2}$ be two real valued random variables with distribution functions $F_{1}$ and $F_{2}$ respectively. If $F_{1}(x)=F_{2}(x)$, $\forall x\in \mathbb{R}$ then $X_{1}$ and $X_{2}$ have the same probability distribution.

I want to prove that the quantile function also characterizes the probability distribution, this fact is stated as a corallary of the above theorem in this book (see Corallary 1.2 on p19):

Corallary: Let $X_{1}$ and $X_{2}$ be two real valued random variables with quantile functions $Q_{1}$ and $Q_{2}$ respectively. If $Q_{1}(u)=Q_{2}(u)$, $\forall u\in\left(0,1\right)$ then $X_{1}$ and $X_{2}$ have the same probability distribution.

The proof in the book is based on the following facts

Fact i: $Q(F(x)) \leq x$.

Fact ii: $F( Q(u) ) \geq u$.

Fact iii: $Q(u) \leq x$ iff $u \leq F(x)$.

Fact iv: $Q(u)$ is nondecreasing.

But I think it is wrong. Assuming $F_{1}(x_0) < F_{2}(x_0)$ for some fixed $x_0$ the author sets out to prove that this leads to a contradiction. Using facts (i) and (iv) he shows $Q_{2}(F_1(x_0)) < Q_{2}(F_2(x_0)) \leq x_0$. Then he applies fact (iii) to obtain $F_{1}(x_0) \leq F_{2}(x_0)$. The author claims that this is a contradiction. But clearly its not and the argument proves nothing.

Am I missing something here? Does anyone know the correct proof to the corallary?

Changed since version discussed in first five comments:

The key line in the proof of Corollary 1.2 in Severini's Elements of Distribution Theory book is

Hence by part (iii) of Theorem 1.8, $F_2(x_0) \ge F_1(x_0)$ so that $F_1(x_0) \lt F_2(x_0)$ is impossible.

As you say, $F_1(x_0) \lt F_2(x_0)$ in fact implies $F_1(x_0) \le F_2(x_0)$, i.e. $F_2(x_0) \ge F_1(x_0)$, so this is not a contradiction.

• I have not miscopied part iii of the theorem. You have got one of the inequalities reversed in the above. But indeed if this was true it would prove the theorem. Commented Oct 10, 2012 at 22:07
• @@Henry: This is what the book says. "$Q(t) \leq x$ iff $F(x)\geq t$". This is what you have written "$Q(t) \leq x$ iff $t \geq F(x)$". You have misread the second inequality. Commented Oct 10, 2012 at 23:02
• @aukie - so I cannot read either; sorry. But the proof of Corallary 1.2 still says $F_{1}(x_0) \ge F_{2}(x_0)$ Commented Oct 10, 2012 at 23:07

I believe this to be the correct proof. It is a direct proof as opposed to the incorrect proof by contradiction given in the book.

$Q_1(F_2(x_0))=Q_2(F_2(x_0)) \leq x_0$ (by part i)

hence

$F_1(x_0) \geq F_2(x_0)$ (by part iii)

Also

$Q_2(F_1(x_0))=Q_1(F_1(x_0)) \leq x_0$ by (by part i)

hence

$F_2(x_0) \geq F_1(x_0)$ (by part iii)

and ofcourse the only way both these conclusions can be true is if $F_1(x_0)=F_2(x_0)$.