To find the marginal density functions for ๐ and ๐, we need to integrate the joint density โ(๐ฅ,๐ฆ) over the appropriate ranges.
The marginal density function of ๐ is found by integrating โ(๐ฅ,๐ฆ) with respect to ๐ฆ over the entire range of ๐ฆ, which in this case is [0,1]:
๐๐(๐ฅ) = โซ[0,1] (๐ฅ+๐ฆ) d๐ฆ
Simplifying the integral, we have:
๐๐(๐ฅ) = ๐ฅโซ[0,1] d๐ฆ + โซ[0,1] ๐ฆ d๐ฆ
= ๐ฅ[y]โยน + [yยฒ/2]โยน
= ๐ฅ(1-0) + (1/2-0)
= ๐ฅ + 1/2
Therefore, the marginal density function of ๐ is ๐๐(๐ฅ) = ๐ฅ + 1/2.
Similarly, the marginal density function of ๐ is found by integrating โ(๐ฅ,๐ฆ) with respect to ๐ฅ over the entire range of ๐ฅ, which is also [0, 1]:
๐๐(๐ฆ) = โซ[0,1] (๐ฅ+๐ฆ) d๐ฅ
Again, simplifying the integral, we have:
๐๐(๐ฆ) = โซ[0,1] ๐ฅ d๐ฅ + ๐ฆโซ[0,1] d๐ฅ
= [๐ฅยฒ/2]โยน + ๐ฆ[x]โยน
= (1/2-0) + ๐ฆ(1-0)
= 1/2 + ๐ฆ
Therefore, the marginal density function of ๐ is ๐๐(๐ฆ) = 1/2 + ๐ฆ.
Let (๐,๐) be a pair of random variables with joint density โ(๐ฅ,๐ฆ)=๐ฅ+๐ฆ over the space [0,1]2.
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