GATE CS 2016 Set 1 Q.14 (CS Q.4): Probability and Statistics

A probability density function on the interval $[a, 1]$ is given by $1/x^2$ and outside this interval the value of the function is zero. The value of $a$ is _____.

Your answer

Official answer and solution

Official answer: 0.5
From: Official GATE 2016 answer key

Solution

Set the integral of the probability density function over its support equal to 1.

For a valid probability density function $f(x)$ , the total area under the curve must equal 1:

\int_{-\infty}^{\infty} f(x) dx = 1

Given $f(x) = \frac{1}{x^2}$ on $[a, 1]$ and zero elsewhere:

\int_a^1 \frac{1}{x^2} dx = 1

Evaluate the definite integral:

\int_a^1 x^{-2} dx = \left[ -\frac{1}{x} \right]_a^1 = -1 - \left(-\frac{1}{a}\right) = \frac{1}{a} - 1

Equate to 1 and solve for $a$ :

\frac{1}{a} - 1 = 1 \implies \frac{1}{a} = 2 \implies a = \frac{1}{2} = 0.5

Final answer

0.5

Common trap

Forgetting the minus sign when integrating $x^{-2}$ , leading to an incorrect sign.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2016 Computer Science and Information Technology (CS) Session 1
Question: 14 of 65, printed as CS Q.4
Section: Computer Science and Information Technology

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