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Part BCSIR NET December 2025a-target-value-can-be-in-the-range-and-still-have-probability-zero-in-a-continuous-distribution

A target value can be in the range and still have probability zero in a continuous distribution

Let {Yn:n≥1Y_{n} : n \ge 1} be a sequence of independent and identically distributed random variables, where Y1Y_{1} ~ Bernoulli(1/2). Define Z=∑n=1∞Z = \sum_{n=1}^\infty 4Yn/5n4Y_{n}/5^{n}. Then, which of the following statements is true?

  1. A.P(Z ≥ 3/5) = 0.5, P(Z = 4/25) = 0✓ Correct
  2. B.P(Z ≥ 3/5) = 0.6, P(Z = 4/25) = 0
  3. C.P(Z ≥ 3/5) = 0.7, P(Z = 4/25) = 0.16
  4. D.P(Z ≥ 3/5) = 0.8, P(Z = 4/25) = 0.16

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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

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: Uncorrelated ⇒ independent

More on this topic

The chapter behind this: Random variables, moments and generating functions — free to read

From Probability › Random variables, distributions, moments, MGF

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