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Part CCSIR NET June 2024max-plus-min-is-the-sum

Max plus min is the sum

Let X1,X2X_{1}, X_{2} denote lifetimes (in years) of 2 components of an electronic system. Let Y1=X1+X2,Y2=max⁡Y_{1} = X_{1} + X_{2}, Y_{2} = \max{X1,X2X_{1}, X_{2}} and Y3=min⁡Y_{3} = \min{X1,X2X_{1}, X_{2}}. Assume that X1X_{1} and X2X_{2} are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

  1. A.P(Y1>2)=2eP(Y_{1} > 2) = 2e⁻1^{1}✓ Correct
  2. B.P(Y2>2)=eP(Y_{2} > 2) = e⁻2^{2}
  3. C.P(Y3>2)=eP(Y_{3} > 2) = e⁻2^{2}✓ Correct
  4. D.Var(Y1+Y2+Y3)=32Var(Y_{1} + Y_{2} + Y_{3}) = 32✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: If X and Y are each normal and uncorrelated then they are independent

More on this topic

The chapter behind this: Joint distributions, transformations and order statistics — free to read

From Probability › Joint distributions, transformations, order statistics

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