Anya has three identical tea cups. She randomly stores each cup in one of three cupboards, where each cupboard nay contain more than one cup, and the choices are uniform and independent. Denote by X the number of tea ups in the rightmost cupboard, and denote by Y the number of non-empty cupboards. 1. Write down the joint distribution of (X, Y).

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Anya has three identical tea cups. She randomly stores each cup in one of three cupboards, where each cupboard
may contain more than one cup, and the choices are uniform and independent. Denote by X the number of tea
cups in the rightmost cupboard, and denote by Y the number of non-empty cupboards.
1. Write down the joint distribution of (X, Y).
2. Compute the marginal expectations E [X] and E [Y].
3. Compute the covariance Cov (X, Y) and the correlation coefficient px.Y .
Transcribed Image Text:1 Anya has three identical tea cups. She randomly stores each cup in one of three cupboards, where each cupboard may contain more than one cup, and the choices are uniform and independent. Denote by X the number of tea cups in the rightmost cupboard, and denote by Y the number of non-empty cupboards. 1. Write down the joint distribution of (X, Y). 2. Compute the marginal expectations E [X] and E [Y]. 3. Compute the covariance Cov (X, Y) and the correlation coefficient px.Y .
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