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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✓
  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.

See pricing

50 are analysed free — try those first.

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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