Math Solutions

[Solution Library] Random variables Y_1 and Y_2 have a joint distribution with constant parameter -1≤ α ≤ 1 : f(y_1,y_2)= (1-α ((1-2e^-y_1)(1-2e^-y_2)))e^-y_1-y_2

Question: Random variables \({{Y}_{1}}\) and \({{Y}_{2}}\) have a joint distribution with constant parameter \(-1\le \alpha \le 1\) :

\[f\left( {{y}_{1}},{{y}_{2}} \right)=\left\{ \begin{aligned} & \left( 1-\alpha \left( \left( 1-2{{e}^{-{{y}_{1}}}} \right)\left( 1-2{{e}^{-{{y}_{2}}}} \right) \right) \right){{e}^{-{{y}_{1}}-{{y}_{2}}}}\text{ for 0}\le {{y}_{1}},0\le {{y}_{2}} \\ & 0\text{ otherwise} \\ \end{aligned} \right.\]
  1. Find the marginal distribution for Y 1 . (4 points)
  2. Find the marginal distribution for Y 2 . (4 points)
  3. For what value of \(\alpha \) are Y 1 and Y 2 independent? (2 points)
  4. Find \(\operatorname{cov}\left( {{Y}_{1}},{{Y}_{2}} \right)\) for all values of \(\alpha \). (4 points)
  5. For what value of \(\alpha \) is \(\operatorname{cov}\left( {{Y}_{1}},{{Y}_{2}} \right)=0\) ? (1 point)
  6. Find \(\operatorname{var}\left[ {{Y}_{1}}-{{Y}_{2}} \right]\). (5 points)

Price: $2.99
Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document


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