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Example Class 5 (sln)

# Example Class 5 (sln) - The University of Hong Kong...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11) Example Class 5 1. Let 𝑋𝑋 and 𝑌𝑌 have the following joint density function: 𝑓𝑓 ( 𝑥𝑥 , 𝑦𝑦 ) = 6 5 ( 𝑥𝑥 2 + 𝑦𝑦 ) 0 , 0 < 𝑥𝑥 , 𝑦𝑦 < 1 , 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (a) Find the marginal pdfs of 𝑋𝑋 and 𝑌𝑌 . 𝑓𝑓 𝑋𝑋 ( 𝑥𝑥 ) = 6 5 ( 𝑥𝑥 2 + 𝑦𝑦 ) 1 0 𝑑𝑑𝑦𝑦 = 6 5 �𝑥𝑥 2 𝑦𝑦 + 𝑦𝑦 2 2 𝑦𝑦 =0 𝑦𝑦 =1 = 6 5 �𝑥𝑥 2 + 1 2 𝑓𝑓 𝑌𝑌 ( 𝑦𝑦 ) = 6 5 ( 𝑥𝑥 2 + 𝑦𝑦 ) 1 0 𝑑𝑑𝑥𝑥 = 6 5 𝑥𝑥 3 3 + 𝑥𝑥𝑦𝑦� 𝑦𝑦 =0 𝑦𝑦 =1 = 6 5 �𝑦𝑦 + 1 3 (b) Find 𝐸𝐸 ( 𝑋𝑋𝑌𝑌 ) , 𝐸𝐸 ( 𝑋𝑋 ) , 𝐸𝐸 ( 𝑌𝑌 ) . 𝐸𝐸 ( 𝑋𝑋𝑌𝑌 ) = � � 6 5 𝑥𝑥𝑦𝑦 ( 𝑥𝑥 2 + 𝑦𝑦 ) 1 0 𝑑𝑑𝑥𝑥 1 0 𝑑𝑑𝑦𝑦 = 6 5 𝑥𝑥 4 𝑦𝑦 4 + 𝑥𝑥 2 𝑦𝑦 2 2 𝑥𝑥 =0 𝑥𝑥 =1 1 0 𝑑𝑑𝑦𝑦 = 6 5 𝑦𝑦 2 8 + 𝑦𝑦 3 6 0 1 = 7 20 𝐸𝐸 ( 𝑋𝑋 ) = 6 5 𝑥𝑥 ( 𝑥𝑥 2 + 𝑦𝑦 ) 1 0 𝑑𝑑𝑥𝑥 = 3 5 𝐸𝐸 ( 𝑌𝑌 ) = 6 5 𝑦𝑦 ( 𝑥𝑥 2 + 𝑦𝑦 ) 1 0 𝑑𝑑𝑦𝑦 = 3 5 (c) Find 𝐶𝐶𝑜𝑜𝐶𝐶 ( 𝑋𝑋 , 𝑌𝑌 ) .

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Example Class 5 (sln) - The University of Hong Kong...

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