Question

If $x$ is a continuous random variable, then $P(a<x<b)=$

• A. $P(a\le x\le b)$
• B. $P(a<x\le b)$
• C. $P(a\le x<b)$
• D. all the three above

Answer 1

Answer: (D)

all the three above

Consider there are $n$ number of points between $a$ and $b$ where $n$ tends to infinite $($given $x$ is a continuous random variable$)$

$P(x=a)=P(x=b)=\frac{1}{n}$ . As $n\to \infty$ , $\frac{1}{n}\to 0$

Therefore $P(x=a)=p(x=b)\to 0$

$⟹P(a<x<b)=P(a\le x\le b)=P(a<x\le b)=P(a\le x<b)$

Therefore the correct option is $D$.