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GATE 2016 CE (CE2) – Question 37

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

Two probability densities are $f(x)=\begin{cases}x/a+1,&-a\le x<0\\-x/a+1,&0\le x\le a\\0,&\text{otherwise}\end{cases}$ and $g(x)=\begin{cases}-x/a,&-a\le x<0\\x/a,&0\le x\le a\\0,&\text{otherwise}.\end{cases}$ Which statement is true?

  1. Same means; same variances
  2. Same means; different variances
  3. Different means; same variances
  4. Different means; different variances

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Correct answer: (B) Same means; different variances

Explanation

Both densities are even, so their first moments vanish and both means are0. Normalisation gives a=1 for the printed heights. The second moments are2∫0¹x²(1−x)dx=1/6 for f and2∫0¹x³dx=1/2 for g. With zero means these are the variances: same means but different variances, **B**.