Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is :
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$A.$ This gives the probability at the point of $b-a$ which is not having any significant w.r.t $a$ and $b.$ $B.$ This gives the difference of the probabilities at $b$ and $a$. Note: This is different from cumulative distribution function $F(b) - F(a).$ Ref: https://en.wikipedia.org/wiki/Cumulative_distribution_function $C.$ This is Probability Density Function. Ref: https://en.wikipedia.org/wiki/Probability_density_function $D.$ This is expected value of continuous random variable. Ref: https://en.wikipedia.org/wiki/Expected_value Answer is $C$.
(B) f(b)−f(a) is not CDF.
Reason: Here function f is pdf not cdf.
Note: If X is Discrete RV then P(a≤X≤b) = F(b) - F(a) ,where F is CDF .Both f and F are diffrent.
@Akash Kanase and @Kapil I think selected answer requires small correction.
@anchitjindal07 For all the $3$ intervals, the probability will be the same as for the closed interval.
$P(a \leq x \leq b ) = \underset{=0}{\underbrace{P(x=a)}}+ \underset{=0}{\underbrace{P(x=b)}} + P(a < x < b )\\P(a \leq x \leq b ) =P(a < x < b ) $
@ Soumya29 Why P(x=a) and P(x=b)= 0