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Part BCSIR NET June 2024chi-squared-two-df-is-exponential

Chi squared two df is exponential

Let X1,X2X_{1}, X_{2} be a random sample from N(0,σ2)N(0, \sigma^{2}) distribution, where σ>0\sigma > 0 and N(μ,σ2)N(\mu, \sigma^{2}) denotes a normal distribution with mean μ\mu and variance σ2\sigma^{2}. Suppose, for some constant c,(c(X12+X22),∞)c, (c(X_{1}^{2} + X_{2}^{2}), \infty) is a confidence interval for variance σ2\sigma^{2} with confidence coefficient 0.95. Then the value of c is equal to

  1. A.−2 ln(0.05)
  2. B.−2 ln(0.95)
  3. C.−1/(2 ln(0.05))✓ Correct
  4. D.−1/(2 ln(0.95))

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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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: The maximum of independent exponentials is exponential

More on this topic

The chapter behind this: The standard distributions — free to read

From Probability › Standard discrete and continuous distributions

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