GATE 2016 EE (EE2) – Question 43

Engineering Mathematics · Probability and Statistics: Random variables · 2 marks · Multiple choice

The density is $f_X(x)=\tfrac32e^{-3x}u(x)+ae^{4x}u(-x)$, with u the unit step. The values of a and P(X≤0), respectively, are

Show answer and explanation

Correct answer: (A) $2,1/2$

Explanation

The positive half-line contains probability $(3/2)\int_0^\infty e^{-3x}dx=1/2$. The negative half-line contains $a\int_{-\infty}^0e^{4x}dx=a/4$.

Normalization requires 1/2+a/4=1, so a=2. Consequently P(X≤0)=a/4=1/2. A single point carries no mass in this density. The pair is **2,1/2, A**.

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