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Part CCSIR NET December 2024hazards-add-only-for-independent-variables

Hazards add only for independent variables

Let X1X_{1} and X2X_{2} be random variables having absolutely continuous distribution functions. Let hi(t)h_{i}(t) denote the hazard function of Xi,i=1,2X_{i}, i = 1, 2. If h1(t)≤h2(t)h_{1}(t) \le h_{2}(t) for all t∈Rt \in \mathbb{R}, then which of the following statements are true?

  1. A.P(X1>1)≥P(X2>1)P(X_{1} > 1) \ge P(X_{2} > 1)✓ Correct
  2. B.P(X1>1)≤P(X2>1/2)P(X_{1} > 1) \le P(X_{2} > 1/2)
  3. C.E(X1)≥E(X2)E(X_{1}) \ge E(X_{2}) provided both the expectations exist.✓ Correct
  4. D.h(t)=h1(t)+h2(t),t∈Rh(t) = h_{1}(t) + h_{2}(t), t \in \mathbb{R}, is the hazard function of the random variable Y = min{X1,X2X_{1}, X_{2}}.

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 dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

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

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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