24 votes 24 votes Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is : $f(b-a)$ $f(b) - f(a)$ $\int\limits_a^b f(x) dx$ $\int\limits_a^b xf (x)dx$ Probability gatecse-2005 probability random-variable easy isro2009 + – gatecse asked Sep 21, 2014 gatecse 11.7k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 27 votes 27 votes $A.$ 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 $C$. Akash Kanase answered Dec 7, 2015 • edited Jun 14, 2018 by Arjun Akash Kanase comment Share Follow See all 10 Comments See all 10 10 Comments reply Nit9 commented Sep 24, 2017 reply Follow Share (b) is not CDF 2 votes 2 votes Shivam Chauhan commented Sep 29, 2017 reply Follow Share @Bikram sir How is b) part CDF 0 votes 0 votes Warrior commented Oct 8, 2017 i edited by Warrior Oct 8, 2017 reply Follow Share (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. 8 votes 8 votes Chhotu commented Oct 14, 2017 reply Follow Share @Akash Kanase and @Kapil I think selected answer requires small correction. 1 votes 1 votes anchitjindal07 commented Nov 26, 2018 reply Follow Share What would be the answer if conditions were: 1. a≤x≤b 2. a≤x<b 3. a<x<b 1 votes 1 votes Soumya29 commented Dec 8, 2018 reply Follow Share @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 ) $ 3 votes 3 votes anchitjindal07 commented Dec 8, 2018 reply Follow Share @ Soumya29 Why P(x=a) and P(x=b)= 0 1 votes 1 votes Verma Ashish commented Oct 19, 2019 reply Follow Share For continuous probability density function f(x) $P(x=a)=\int_a^af(x)dx=0$ Probability at discrete point=0 So here random variables should be defined as continuous not discrete 3 votes 3 votes shishir__roy commented Sep 11, 2022 reply Follow Share For option (A). $f(b-a)$ gives the probability that the random variable $x$ takes a value near point $(b-a)$ Similarly, for option (B). $f(b) – f(a)$ gives the difference of probabilities that random variable $x$ takes a value near point $b$ and point $a$ 0 votes 0 votes JAINchiNMay commented Sep 25, 2022 reply Follow Share In question f(x) is given probability density function and again the option C is probaility density function .how? 0 votes 0 votes Please log in or register to add a comment.
16 votes 16 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 Bhagirathi answered Sep 21, 2014 Bhagirathi comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes f(x) be the continuous probability density function of random variable X. Then the probablity be area of the corresponding curve i.e., varunrajarathnam answered Nov 13, 2020 varunrajarathnam comment Share Follow See all 0 reply Please log in or register to add a comment.