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Part BCSIR NET December 2023

Random variables, distributions, moments, MGF

Suppose X ~ Poisson(3/4). Then which of the following statements is true?

A.P(X > 9) ≥ 11/12
B.P(X < 9) ≥ 11/12✓
C. $E (X - 3/4)^{2} \geq 11/12$
D.(11/9)X ~ Poisson(11/12)

Solution

Markov: P(X ≥ 9) ≤ E[X]/9 = 1/12, so P(X < 9) ≥ 11/12. The variance is 3/4 < 11/12, and a scaled Poisson is not Poisson.

Related counterexample: Uncorrelated ⇒ independent

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From Probability › Random variables, distributions, moments, MGF

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