GATE 2022 BT – Question 31
A random variable has density $p(x)=2e^{-2x}$. The probability $P(1\le X\le2)$ (two decimal places) is ___________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.11 to 0.13
Explanation
For a probability density $p(x)=2e^{-2x}$ (an exponential distribution),
$$P(1\le X\le2)=\int_1^2 2e^{-2x}\,dx=\left[-e^{-2x}\right]_1^2=e^{-2}-e^{-4}.$$
**Numbers.** $e^{-2}=0.1353$ and $e^{-4}=0.0183$:
$$P=0.1353-0.0183=0.117\approx\mathbf{0.12}.$$