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Example Class 3 (sln)

Example Class 3 (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 3 Solution 1. Suppose random variables 𝑋𝑋 and 𝑌𝑌 have the joint pdf 𝑓𝑓 ( 𝑥𝑥 , 𝑦𝑦 ) = 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 ) for 𝑥𝑥 , 𝑦𝑦 ∈ [0,1] . (a) Find 𝑃𝑃 ( 𝑌𝑌 ≥ 2 𝑋𝑋 ) . 𝑃𝑃 ( 𝑌𝑌 ≥ 2 𝑋𝑋 ) = 3 ( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 ) 1 2 𝑥𝑥 𝑑𝑑𝑦𝑦 1 2 0 𝑑𝑑𝑥𝑥 = 3 𝑥𝑥𝑦𝑦 − 𝑥𝑥𝑦𝑦 2 + 𝑦𝑦 3 3 𝑦𝑦 =2 𝑥𝑥 𝑦𝑦 =1 1 2 0 𝑑𝑑𝑥𝑥 = 3 ( 𝑥𝑥 − 𝑥𝑥 + 1 3 1 2 0 2 𝑥𝑥 2 + 4 𝑥𝑥 3 + 8 𝑥𝑥 3 3 ) 𝑑𝑑𝑥𝑥 = 3 𝑥𝑥 3 2 𝑥𝑥 3 3 + 𝑥𝑥 4 + 8 𝑥𝑥 4 12 0 1 2 = 5 16 (b) Find the marginal pdfs of 𝑋𝑋 and 𝑌𝑌 respectively. 𝑓𝑓 𝑋𝑋 ( 𝑥𝑥 ) = 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 ) 1 0 𝑑𝑑𝑦𝑦 = 3 �𝑥𝑥𝑦𝑦 − 𝑥𝑥𝑦𝑦 2 + 𝑦𝑦 3 3 0 1 = 1 𝑓𝑓 𝑌𝑌 ( 𝑦𝑦 ) = 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 ) 1 0 𝑑𝑑𝑥𝑥 = 3 𝑥𝑥 2 2 − 𝑥𝑥 2 𝑦𝑦 + 𝑥𝑥𝑦𝑦 2 0 1 = 3 𝑦𝑦 2 3 𝑦𝑦 + 3 2 (c) Are 𝑋𝑋 and 𝑌𝑌 independent? 𝑓𝑓 𝑋𝑋 ( 𝑥𝑥 ) 𝑓𝑓 𝑌𝑌 ( 𝑦𝑦 ) ≠ 𝑓𝑓 ( 𝑥𝑥 , 𝑦𝑦 ) Not Independent (d) Find 𝑓𝑓 𝑋𝑋 | 𝑌𝑌 ( 𝑥𝑥 | 𝑦𝑦 ) . 𝑓𝑓 𝑋𝑋 | 𝑌𝑌 ( 𝑥𝑥 | 𝑦𝑦 ) = 𝑓𝑓 ( 𝑥𝑥 , 𝑦𝑦 ) 𝑓𝑓 𝑌𝑌 ( 𝑦𝑦 ) = 3( 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 ) 3 𝑦𝑦 2 3 𝑦𝑦 + 3 2
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= 𝑥𝑥 − 2 𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 𝑦𝑦 2 − 𝑦𝑦 + 1 2 2. Suppose 𝑋𝑋 and 𝑌𝑌 are discrete random variables with a joint pmf as following: X -1 0 1 -1 1/6 1/9 1/9 Y 0 1/9 0 c 1 1/18 1/9 1/6 (a) Find the value of c.
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