Example Class 3 - The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics Foundations of

Example Class 3 - The University of Hong Kong Department of...

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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 1.Suppose random variables 𝑋𝑋and 𝑌𝑌have the joint pdf 𝑓𝑓(𝑥𝑥,𝑦𝑦) = 3(𝑥𝑥 −2𝑥𝑥𝑦𝑦+𝑦𝑦2)for 𝑥𝑥,𝑦𝑦 ∈[0,1](a) Find 𝑃𝑃(𝑌𝑌 ≥2𝑋𝑋)(b) Find the marginal pdfs of 𝑋𝑋and 𝑌𝑌respectively. (c) Are 𝑋𝑋and 𝑌𝑌independent? (d) Find 𝑓𝑓𝑋𝑋|𝑌𝑌(𝑥𝑥|𝑦𝑦)2.Suppose 𝑋𝑋and 𝑌𝑌are discrete random variables with a joint pmf as following: X -1 0 -1 1/6 1/9 Y 0 1/9 1 1/18 1/9 (a)Find the value of c. (b)Find the marginal pmfs of X and Y respectively. (c)Are X and Y independent? (d)Find 𝑃𝑃(𝑌𝑌> 0.5),𝑃𝑃(𝑋𝑋>0.5,𝑌𝑌> 0.5)and 𝑃𝑃(𝑋𝑋>0.5|𝑌𝑌= 1)3.Let the joint pdf of 𝑋𝑋and 𝑌𝑌𝑓𝑓(𝑥𝑥,𝑦𝑦) =𝑥𝑥𝑒𝑒(𝑥𝑥+𝑦𝑦)for 𝑥𝑥,𝑦𝑦> 0(a) Show that 𝑋𝑋and 𝑌𝑌are independent. (b) Find 𝑃𝑃(𝑋𝑋 ≤ 𝑎𝑎,𝑌𝑌 ≤ 𝑏𝑏)for constants 𝑎𝑎and 𝑏𝑏(c) Find 𝑃𝑃(𝑋𝑋 ≤ 𝑎𝑎)(d) Find 𝑃𝑃(𝑋𝑋+𝑌𝑌 ≤ 𝑎𝑎)4.Let 𝑋𝑋and 𝑌𝑌be two discrete random variables with a joint pmf 𝑝𝑝(𝑥𝑥,𝑦𝑦) =𝑥𝑥𝑦𝑦𝑒𝑒−𝑥𝑥𝑁𝑁×𝑦𝑦!where 𝑁𝑁is a positive integer, 𝑥𝑥= 1,2, … ,𝑁𝑁and 𝑦𝑦= 1,2, … . . . 1 1/9 0 c 1/6 . be . . . .

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