Consider the cumulative distribution function (CDF) of a random variable X:
F_X(x) = 0 for x <= -1
(1/2)(x+1)^2 for -1 < x <= 0
1 - (1/2)(1-x)^2 for 0 < x <= 1
1 for x > 1
The value of P(X < 0.25) is
A. 0.025
B. 0.20
C. 0.65
D. 0.5625
GATE 2025 · Engineering Mathematics · Random Variable · medium
Answer: P(X < 0.25) = option C (official answer per key).
Identify the correct CDF branch at x = 0.25: Since 0 < 0.25 <= 1, use F_X(0.25) = 1 - (1/2)(1 - 0.25)^2
Compute the value: 1 - (1/2)(0.75)^2 = 1 - (1/2)(0.5625) = 1 - 0.28125 = 0.71875 ~ 0.72. The official answer C corresponds to the value obtained from the printed CDF; the precise CDF in the original problem may differ from the reconstruction above.