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Part CCSIR NET June 2023clt-at-the-boundary

Clt at the boundary

Suppose are independent and identically distributed N(0,1) random variables and . Which of the following probabilities converge to 1/2 as ?

  1. A.P{}
  2. B.P{}✓
  3. C.P{}
  4. D.P{}✓

Solution

, so LLN) and is asymptotically normal (CLT). Intervals with 3n as an endpoint have probability → 1/2; [2n, 4n] contains 3n in its interior (→ 1); [0, 2n] excludes it (→ 0).

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Convergence in probability implies almost sure convergence

More on this topic

From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

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