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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. $\int\limits_a^b xf (x)dx$

$A.$ 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 $C$.

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
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 $a$

In question f(x) is given probability density function and again the option is probaility density function .how?

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

f(x) be the continuous probability density function of random variable X.
Then the probablity be area of the corresponding curve i.e.,

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