The GATE Grind

GATE 2025 DA – Question 41

Probability and Statistics · Random variables, discrete and continuous distributions · 2 marks · Multiple choice

For $x \in \mathbb{R}$, the floor function is denoted by $f(x) = \lfloor x \rfloor$ and defined as follows

$\lfloor x \rfloor = k$, $k \le x < k + 1$,

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

  1. 0.1
  2. 0.01
  3. 0.5
  4. 0.434

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Show answer and explanation

Correct answer: (A) 0.1

Explanation

The rate is $\lambda = \ln 10$, so $P(X \ge x) = e^{-x\ln 10} = 10^{-x}$. Then $P(Y = \ell) = P(\ell \le X < \ell + 1) = 10^{-\ell} - 10^{-(\ell+1)} = 10^{-\ell}(1 - 10^{-1})$. This has the form $q^\ell(1 - q)$ with $q = 0.1$.