題組內容

3. Given λ, a fixed positive constant, define the following function:

$$f(x) = \frac{1}{2} \lambda e^{-\lambda x}, \text{ for } x \ge 0; \quad f(x) = \frac{1}{2} \lambda e^{\lambda x}, \text{ for } x < 0.$$



a. (10%) If the probability density function of the random variable \( X \) is \( f(x) \), find \( P(X < t) \) for all possible values of \( t \).