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HW8_solutions

# HW8_solutions - HW#8 Solutions AMS 311 Probability...

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HW#8 Solutions AMS 311 Probability Theory (Spring 2011) Part 1 (10 points): PRINT your name and StonyBrook ID ( Last, First, #ID ) on the top on your work sheet. Staple your HW if more than 1 pages. Part 2: Problems (90 points) Write down your work step by step to get full score. You do not need to evaluate arithmetic expressions or integrals, if they are fully specified. For example, you may leave in this form. (1) (24 points) (a) Var(2X)=4Var(X)=3, so Var(X)=3/4. So, E(3X)=Var(X/2)=Var(X)/4=3/16, so EX=1/16 (i) EX 2 =Var(X)+(EX) 2 =193/256 E[(2 + X) 2 ]=E(X 2 +4X+4)=EX 2 +4EX+4=1281/256 (ii)Var(4+3X)=9Var(X)=27/4 (b) We know Var(W)=7 E((X − W)(X + W))=E(X 2 -W 2 )=EX 2 -EW 2 =100. E(3W)=30, so EW=10. E(X+W)=EX+EW=12, so EX=2. EW 2 =VarW+(EW) 2 =7+100=107, so EX 2 =207 (i) Var(X)=EX 2 -(EX) 2 =203 (ii) cov(X 3 , W 2 )=0 (due to independence, E(X 3 W 2 )=E(X 3 )E(W 2 ).) (2). (13 points) You don’t have to calculate all the integrations. Homework is due at the beginning of class on its due date. You must be in class and on time to submit your homework. No homework will be accepted via email. No late homework will be accepted. Your

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HW8_solutions - HW#8 Solutions AMS 311 Probability...

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