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✓ Correct
C. $E(X - 3/4)^{2} \ge 11/12$
D.(11/9)X ~ Poisson(11/12)

The idea, in plain words

A variable with a small average very rarely takes a large value — that alone bounds the tail, without needing the exact distribution. Compare that bound with the spread, and watch out for the option that assumes scaling a count keeps it a count.

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.

Why each option is right or wrong

Checked against 3 symbolic computations

A.WrongExecution slip
False, and it is (B) with the inequality pointing the wrong way.

A Poisson variable with mean 3/4 is concentrated near 0 — Markov's inequality gives P(X ≥ 9) ≤ (3/4)/9 = 1/12 (verified)

so the upper tail is tiny, not overwhelming.
B.Correct
True.

X is non-negative

so Markov applies: P(X ≥ 9) ≤ E[X]/9 = (3/4)/9 = 1/12 (verified).

Taking complements, P(X < 9) ≥ 1 − 1/12 = 11/12 (verified).

The trigger: a bound of the form 'probability ≥ 1 − something' on a non-negative variable is almost always Markov on the complement.
C.WrongExecution slip
False.

$E(X - 3/4)^{2}$ is the variance, and for a Poisson distribution the variance equals the mean

so it is 3/4 = 0.75.

That is less than 11/12 ≈ 0.917 — SymPy gives the difference as −1/6

confirming the inequality fails.
D.WrongProperty not inherited
False.

Scaling a Poisson variable does not give a Poisson variable: (11/9)X takes values 0, 11/9, 22/9, … which are not non-negative integers

so it cannot have any Poisson distribution.

The Poisson family is closed under addition of independent variables, not under multiplication by constants.

3 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.

Last revised 11 Sept 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

Random variables, distributions, moments, MGF

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