A probability density function is of the form $${\text{p}}\left( {\text{x}} \right) = {\text{K}}{{\text{e}}^{ – \alpha \left| x \right|}},\,{\text{x}} \in \left( { – \infty ,\,\infty } \right).$$ The value of K is A. 0.5 B. 1 C. 0.5$$\alpha $$ D. $$\alpha $$

The correct answer is B. 1.

A probability density function (PDF) is a function that gives the probability of a random variable taking on a value within a given interval. The PDF is always non-negative and integrates to 1 over the entire real line.

In this case, the PDF is given by

$$p(x) = K e^{-\alpha |x|}, \quad x \in (-\infty, \infty)$$

The value of $K$ can be found by integrating the PDF over the entire real line. This gives

$$1 = \int_{-\infty}^{\infty} p(x) dx = K \int_{-\infty}^{\infty} e^{-\alpha |x|} dx = K \left[ \frac{1}{2 \alpha} e^{-\alpha |x|} \right]_{-\infty}^{\infty} = K$$

Therefore, $K = 1$.

The other options are incorrect because they do not satisfy the condition that the PDF integrates to 1 over the entire real line.