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+10 votes

Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is :

  1. $f(b-a)$
  2. $f(b) - f(a)$
  3. $\int\limits_a^b f(x) dx$
  4. $\int\limits_a^b xf (x)dx$
asked in Probability by Boss (18.3k points) | 2.1k views

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2 Answers

+11 votes
Best answer

$A.$ 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:

$C.$ This is Probability Density Function. Ref:

$D.$ This is expected value of continuous random variable. Ref:
Answer is $C$.

answered by Boss (42.9k points)
edited by
(b) is not CDF
@Bikram sir How is b) part CDF

(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. 

+13 votes
C should be used if prob density function is given B should be used if prob distribution function is given D must be used to calculate expectation when pdf is given
answered by Boss (14.3k points)

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