GATE 2017 CE (CE1) – Question 36

Engineering Mathematics · Probability and Statistics: Random variables, Poisson and Normal distributions, linear regression · 2 marks · Multiple choice

For $f(x)=a+bx$ on $0\leq x\leq1$ to be a valid probability density, which statement is correct?

$a=1,b=4$
$a=0.5,b=1$
$a=0,b=1$
$a=1,b=-1$

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Show answer and explanation

Correct answer: (B) $a=0.5,b=1$

Explanation

A density must be nonnegative and integrate to 1. Here $\int_0^1(a+bx)dx=a+b/2$. For A, C and D this equals 3, 0.5 and 0.5 respectively, so they fail normalization. B gives $0.5+1/2=1$, and $f(x)=0.5+x$ is positive throughout the interval. Thus B is valid.

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