Multivariate probability distributions

Question

Assignment:

Multivariate Probability Distributions

Y₁ & Y₂  denotes the proportions of time that employee I and II actually spent working on their assigned tasks during a workday.  The joint density of Y₁ & Y₂  is given by:

f(y₁ ,y₂)  =  y₁ + y₂ , 0 ≤ y₁ ≤ 1    ,  0 ≤ y₂ ≤ 1
f(y₁ ,y₂) =  0,  elsewhere

Employee I has a higher productivity rating than employee II, and a measure of the total productivity of the pair of employees is 3o Y₁ + 25 Y₂ .  Find the expected value of this measure of productivity.

Provide complete and step by step solution for the question and show calculations and use formulas.