Part A

$2.$ Over one period, an investment in TiVo stock offers the following possible returns with corre$-$sponding probabilities: a low return of $8\mathrm{.}0\%$ with a $20\%$ chance, a medium return of $14\mathrm{.}0\%$ with a $50\%$ chance, and a high return of $17\mathrm{.}0\%$ with a $30\%$ chance. Over the same period, an invest$-$ment in Microsoft stock offers the following possible returns with corresponding probabilities: $6\mathrm{.}0\%$ with a $20\%$ chance, $11\mathrm{.}0\%$ with a $50\%$ chance, and $12\mathrm{.}0\%$ with a $30\%$ chance. The returns on the two investments are such that only three states are possible: low returns, medium returns, and high returns.

Suppose the betas for TiVo$$s stock and Microsoft$$s stock are $2\mathrm{.}1$ and $1\mathrm{.}2,$ respectively.

$(1)$ Calculate the beta for portfolio X$.$

$(2)$ Calculate the beta for portfolio Y$.$