Question

For \[ f_X(x) = \frac{1}{\pi} \left(\frac{q}{e^x + e^{-x}}\right) \] to be a valid probability distribution function of a random variable X, the value of q is

• A. 2
• B. $\pi$
• C. 4
• D. $-\pi$

Hint:

For any valid probability distribution function, the total area under the curve (integral) must equal 1.

Answer 1

The Correct Option is A

For a probability distribution function to be valid, the integral of the function over all possible values must be 1. That is, \[ \int_{-\infty}^{\infty} f_X(x) \, dx = 1. \] Given the function \[ f_X(x) = \frac{1}{\pi} \left( \frac{q}{e^x + e^{-x}} \right), \] we can simplify \( e^x + e^{-x} \) as \( 2\cosh(x) \), where \( \cosh(x) \) is the hyperbolic cosine function. Thus, the integral becomes: \[ \int_{-\infty}^{\infty} \frac{q}{\pi} \cdot \frac{1}{2 \cosh(x)} \, dx. \] The integral of \( \frac{1}{\cosh(x)} \) over all \( x \) is a known result and equals \( \pi \). Therefore, \[ \frac{q}{2\pi} \cdot \pi = 1 $\Rightarrow$ \frac{q}{2} = 1 $\Rightarrow$ q = 2. \] Thus, the correct value of \( q \) is Option (A).