GATE 2025 · session-5
For \( x \in \mathbb{R} \), the floor function is denoted by \( f(x) = \lfloor x \rfloor \) and defined as follows
\[ \lfloor x \rfloor = k, \quad k \leq x < k + 1, \]
where \( k \) is an integer. Let \( Y = |X| \), where \( X \) is an exponentially distributed random variable with mean \( \frac{1}{\ln 10} \), where \( \ln \) denotes natural logarithm. For any positive integer \( \ell \), one can write the probability of the event \( Y = \ell \) as follows:
\[ P(Y = \ell) = q^\ell (1 - q) \]
The value of \( q \) is: