Question

A random variable $x$ assumes values which are numbers of the form $\frac{n}{n+1}$ and $\frac{n+1}{n}$, where

$n=1,2,3,...$. lf $P(x=\frac{n}{n+1})=P(x=\frac{n+1}{n})={\left(\frac{1}{2}\right)}^{n+1}$, then

• A. $P(x<1)=P(x>1)$
• B. $P(1/2<x<1)<P(x>1)$
• C. $P(x>3/2)<P(x<1)$
• D. $P(x>3/2)=0$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $P(x<1)=P(x>1)$
B $P(1/2<x<1)<P(x>1)$
C $P(x>3/2)<P(x<1)$

The numbers of the form $\frac{n}{n+1}$ are less than 1 and $\frac{n+1}{n}$ are greater than 1.
$P(x<1)=P\left(\frac{1}{2}\right)+P\left(\frac{2}{3}\right)+...$
$\Rightarrow P(x<1)=\frac{1}{4}+\frac{1}{8}+...$
$\Rightarrow P(x<1)=\frac{\frac{1}{4}}{1-\frac{1}{2}}$
$\Rightarrow P(x<1)=\frac{1}{2}$
$P(x>1)=\frac{1}{2}$
$P(\frac{1}{2}<x<1)=P\left(\frac{2}{3}\right)+P\left(\frac{3}{4}\right)+...$

$P(\frac{1}{2}<x<1)=\frac{1}{8}+\frac{1}{16}+...$

$P(\frac{1}{2}<x<1)=\frac{1}{4}$
$P(x>\frac{3}{2})=P(x=2)$
$P(x>\frac{3}{2})=\frac{1}{4}$
$P(x>\frac{3}{2})=\frac{1}{4}$]

$\therefore$$P(x<1)=P(x>1)$

$P(1/2<x<1)<P(x>1)$

$P(x>3/2)<P(x<1)$