For \(x \in \mathbb{R}\), the floor function is denoted by \(f(x) = \lfloor x \rfloor\) and defined as follows
\[\lfloor x \rfloor = k, \;\; k \le x \lt k + 1,\]
where \(k\) is an integer. Let \(Y = \lfloor X \rfloor\), 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
Answer
(A)
Explanation
A mean of \(\frac{1}{\ln 10}\) means the rate is \(\lambda = \ln 10\), so \(P(X \ge t) = e^{-t \ln 10} = 10^{-t}\) for \(t \ge 0\).
\(Y = \ell\) happens exactly when \(\ell \le X \lt \ell + 1\), so
\[P(Y = \ell) = 10^{-\ell} - 10^{-(\ell + 1)} = (0.1)^{\ell}\,(1 - 0.1).\]
This has the form \(q^{\ell}(1 - q)\) with \(q = 0.1\), so the floor of an exponential variable is geometric. Option D, \(0.434 \approx \frac{1}{\ln 10}\), is the mean of \(X\), not \(q\).
Answer: A