GATE 2023 ST – Question 53
X has density e^(−x) for x≥0,zero otherwise. Which hold? U(a,b) denotes the uniform law on (a,b).
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Correct answer: (C) 2e^(−X)−1∼U(−1,1); (D) For Y=⌊X⌋,P(Y=k)=(1−e⁻¹)e^(−k),k=0,1,2,…
Explanation
The probability integral transform gives 1−e^(−X)∼U(0,1),and its complement e^(−X) is also U(0,1). The affine transformation in C is therefore U(−1,1). For D,P(k≤X<k+1)=e^(−k)−e^(−k−1). Hence **C,D**.