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Distinguish a reflected distribution from dependent realised throws

A cubical die has three faces marked 1, two faces marked 2 and one face marked 3. Find the expectation and variance of its score in one throw.

Deduce, or find otherwise, the expectation and variance for a second cubical die with one face marked 1, two marked 2 and three marked 3.

Two dice of the first type and one of the second are thrown together; X is their total score. Let μ=E(X)\mu=E(X) and σ2=Var⁡(X)\sigma^2=\operatorname{Var}(X). Show that

(i) σ2=5/3\sigma^2=5/3,

(ii) P(∣X−μ∣>2σ)=1/18P(|X-\mu|>2\sigma)=1/18.

[J1983/I/11]

CJC · Discussion · Question 6

Reflecting the face-label distribution is useful for one-die moments; use independent copies for the three-die total.

Read the worked solution