Question

Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

• A. $F\left(x\right)-G\left(x\right)\le 0$
• B. $F\left(x\right)-G\left(x\right)\ge 0$
• C. $F\left(x\right)-G\left(x\right)x\le 0$
• D. $F\left(x\right)-G\left(x\right).x\ge 0$

Answer 1

The correct option is D $F\left(x\right)-G\left(x\right).x\ge 0$

By definition of c.d.f we have

$F\left(x\right)=P[u\le x]$

& $G\left(x\right)=P[2v\le x]$

$=P(v\le \frac{x}{2})$

Case I: When x > 0; F(x) > G(x)

Case II: when x < 0; F(x) < G(x) So in both cases
$\left[f\right(x)-G(x\left)\right]x\ge 0$