Question

Let the probability density function of a random variable \( x \) be given as \[ f(x) = ae^{-2|x|} \] The value of ‘a’ is ________.

Hint:

To find the constant \( a \) in a probability density function, integrate the function over the entire range and set the result equal to 1.

Answer 1

Correct Answer: 0.99

The probability density function must satisfy the condition that the total probability is equal to 1. This is expressed as: \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \] Substituting the given function \( f(x) = ae^{-2|x|} \) into the equation: \[ \int_{-\infty}^{\infty} ae^{-2|x|} \, dx = 1 \] We split the integral into two parts because of the absolute value: \[ \int_{-\infty}^{0} ae^{2x} \, dx + \int_{0}^{\infty} ae^{-2x} \, dx = 1 \] The integrals can be solved as: \[ \int_{-\infty}^{0} ae^{2x} \, dx = \frac{a}{2}, \quad \int_{0}^{\infty} ae^{-2x} \, dx = \frac{a}{2} \] Thus: \[ \frac{a}{2} + \frac{a}{2} = 1 \] Solving for \( a \): \[ a = 1 \] Thus, the value of \( a \) is \( \boxed{1} \).