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Part CCSIR NET December 2024the-sample-is-of-systems-not-components

The sample is of systems not components

Consider a manufacturing unit producing three-component series systems, where component's lifetimes are independent and identically distributed random variables with hazard function h(x)=λh(x) = \lambda if x > 0, and 0 otherwise. Here, λ>0\lambda > 0 is an unknown parameter. Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample of size 3 on lifetimes of systems produced by the manufacturing unit. Let θ=Pλ(X1>1/2)\theta = P_\lambda(X_{1} > 1/2). If θ\thetâ is the maximum likelihood estimate of θ\theta based on the realization x1=1/2,x2=1/6,x3=1/3x_{1} = 1/2, x_{2} = 1/6, x_{3} = 1/3 of the random sample, then which of the following statements are true?

  1. A.θ\thetâ ∈ (0.0, 0.2)
  2. B.θ\thetâ ∈ (0.1, 0.3)✓ Correct
  3. C.θ\thetâ ∈ (0.2, 0.4)✓ Correct
  4. D.θ\thetâ ∈ (0.3, 0.5)

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: The MLE is unbiased

More on this topic

The chapter behind this: Maximum likelihood and the method of moments — free to read

From Estimation › MLE and method of moments

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