Inequality involving joint cumulative and marginal distributions

Question

We need to establish the given inequality : $$F_X(x) + F_Y(y) - 1 \leq F_{X,Y}(x,y) \leq \sqrt{F_X(x) F_Y(y)}$$ where $X$ and $Y$ are any random variables.

First I tried the R.H.S :

I started working on p.d.f's ( probability density functions first ).

$ f_{X|Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)}\implies \dfrac{f_{X|Y}(x|y)}{f_X(x)} = \dfrac{f_{X,Y}(x,y)}{f_Y(y)f_X(x)}$

But I don't think that's gonna work . Can anyone suggest some other way ?

Answer 1

Hint for the right hand side:

$F_X(x) = P(X < x) \ge P(X < x \wedge Y < y) = F_{X, Y}(x, y)$.

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Hint for the left hand side:

The following figure shows the $XY$ plane, with the horizontal and vertical lines intersecting at $(x, y)$.

The region $C$ is the area where $X < x \wedge Y < y$ - you need to integrate over it to obtain $F_{X, Y}(x, y)$. Over which regions do you need to integrate to obtain $F_X(x)$, $F_Y(y)$, and $1$?

Answer 2

It is just the elementary inequality $$P(A)+P(B)-1\le P(A\cap B)\le \sqrt{P(A)P(B)}$$ for events $A=\{X\le x\}$ and $B=\{Y\le y\}$. There is no need to go into distributions.

Let $I_A$ be the indicator of $A$, i.e. $I_A=1$ if $A$ occurs and $I_A=0$ if $A$ does not occur.

Then by Cauchy-Schwarz inequality,

$$\left(E\left[I_AI_B\right]\right)^2 \le E\left[I_A^2\right]E\left[I_B^2\right] \implies (P(A\cap B))^2\le P(A)P(B)$$

And $P(A^c\cup B^c)\le P(A^c)+P(B^c)=1-P(A)+1-P(B)$ implies $$P(A\cap B)=1-P(A^c\cup B^c)\ge P(A)+P(B)-1$$