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GATE DA 2025 Set 1 — Question 41

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MCQ+2 / -0.67MediumExponentialContinuous Random VariablesProbability & StatisticsPMFDiscrete Random Variables

Probability & Statistics → Continuous Random Variables → Exponential

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Question

For x∈Rx \in \mathbb{R}, the floor function is denoted by f(x)=⌊x⌋f(x) = \lfloor x \rfloor and defined as follows⌊x⌋=k,k≤x<k+1,\lfloor x \rfloor = k, \quad k \le x < k + 1,where kk is an integer. Let Y=⌊X⌋Y = \lfloor X \rfloor, where XX is an exponentially distributed random variable with mean 1ln⁡10\frac{1}{\ln 10}, where ln⁡\ln denotes natural logarithm. For any positive integer ℓ\ell, one can write the probability of the event Y=ℓY = \ell as followsP(Y=ℓ)=qℓ(1−q)P(Y = \ell) = q^\ell (1 - q)The value of qq is
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