GATE 2024 ME – Question 44
Let $X$ be a continuous random variable defined on $[0, 1]$ such that its probability density function $f(x) = 1$ for $0 \le x \le 1$ and 0 otherwise. Let $Y = \log_e(X + 1)$. Then the expected value of $Y$ is_____ (*rounded off to 2 decimal places*).
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Correct answer: 0.38 to 0.40
Explanation
$E[Y] = \int_0^1 \ln(x + 1)\,dx = \left[(x + 1)\ln(x + 1) - (x + 1)\right]_0^1 = (2\ln 2 - 2) - (0 - 1) = 2\ln 2 - 1 = 0.386$, which is 0.39.