Question

The function p(x) is given by p(x) = A/x^μ where A and μ are constants with μ > 1 and 1 < x < ∞ and p (x) = 0 for -∞ < x < 1. For p(x) to be a probability density function, the value of A should be equal to
1. μ - 1
2. μ + 1
3. 1/ (μ - 1)
4. 1/ (μ + 1)

Answer 1

Correct Answer - Option 1 : μ - 1

\(p\left( x \right) = \frac{A}{{{x^\mu }}},1 \le x < \infty\)

= 0, - ∞ < x < 1

p(x) to be a probability density function

\(\Rightarrow \mathop \smallint \limits_{ - \infty }^\infty p\left( x \right)dx = 1\)

\(\Rightarrow \mathop \smallint \limits_1^\infty \frac{A}{{{x^\mu }}}dx = 1\)

\(\Rightarrow \left[ {A\frac{{{x^{ - \mu + 1}}}}{{ - \mu + 1}}} \right]_1^\infty = 1\)

\(\Rightarrow \frac{A}{{1 - \mu }}\left[ {{{\left( \infty \right)}^{1 - \mu }} - {{\left( 1 \right)}^{ - \mu + 1}}} \right] = 1\)

\(\Rightarrow \frac{A}{{1 - \mu }}\left[ {0 - 1} \right] = 1\)

⇒ A = μ – 1