GATE CS 2005 Q12 — Probability: Random Variables and Distributions

For a continuous random variable $X$ with a probability density function (PDF) $f(x)$, the probability that $X$ takes a value in the interval $(a, b]$ is defined as the area under the PDF curve between $x = a$ and $x = b$.

Mathematically, this is expressed as:

Analysis of Options:

• Option A ($f(b-a)$): This represents the value of the density function at the point $b-a$, which is not a probability.
• Option B ($f(b) - f(a)$): This is the difference in the density values at $b$ and $a$. Note that if $F(x)$ were the Cumulative Distribution Function (CDF), then $P(a < X \leq b) = F(b) - F(a)$, but here $f(x)$ is the PDF.
• Option C ($\int_{a}^{b} f(x) \, dx$): This is the correct definition of the probability for a continuous random variable.
• Option D ($\int_{a}^{b} x f(x) \, dx$): This expression is used to compute the partial expectation of $X$ over the interval $[a, b]$, not the probability.

Therefore, the correct option is C.