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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 Spring '08
 Tucker,A
 Probability theory, lowest homework grade

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