Q.31 The probability density function of a random variable X is
p(x) = 2e^-2x. The probability P(1 ≤ X ≤ 2) (rounded off to two decimal places) is ________.

The probability density function p(x) = 2e^−2x defines an exponential distribution with rate parameter λ = 2, valid for x ≥ 0.
Computing P(1 ≤ X ≤ 2) requires integrating this PDF over the interval, yielding
0.12 when rounded to two decimal places.

PDF Verification

This PDF integrates to 1 over [0, ∞), confirming validity:

∫₀^∞ 2e^−2x dx = 1.

The exponential form matches standard f(x) = λe^−λx with λ = 2.

Step-by-Step Calculation

P(1 ≤ X ≤ 2) = ∫₁² 2e^−2x dx

The antiderivative of 2e^−2x is −e^−2x.

Evaluate:

[−e⁻⁴] − [−e⁻²] = e⁻² − e⁻⁴

Numerically:

• e⁻² ≈ 0.135335
• e⁻⁴ ≈ 0.0183156
• Difference = 0.117019 ≈ 0.12

Common Options Explanation

• 0.06: Incorrect; confuses antiderivative factor
• 0.12: Correct answer
• 0.43: Miscomputes 1 − e⁻²
• 0.86: Equal to P(X ≤ 1)