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Answered: - need solution of 10-6 please help Joe Zilch has found that the
?need solution of 10-6 please help?Joe Zilch has found that the hours spent working (W) and sleeping (S) in preparation foran exam are random variables described byfS,Ws,w( ) =e?s,0?w?1 and0?s? ?0,elsewhere$%It turns out that Joe?s grade (G) on the exam is helped more by his working than by hissleeping, and that his grade is described byG=S+2Wa). Find and make a labeled sketch of the probability density functionfGg( )for all values ofG.b). Find the conditional density ofGgiven that Joe?s gradeGwas less than 1.
EE 503
Problem Set #10
Due Wednesday, March 30, 2016
Spring 2016
Sawchuk
Wednesday, March 30
- Lecture 19, including review for Midterm 2
- Problem Set #10 due
Friday, April 1
- Discussion session 8:00 am - 8:50 am
- 4:00 pm - final deadline to submit Problem Set #10 in EEB 404; solutions will be posted at that
time
Monday, April 4 - Midterm 2
- 10:00 am-11:50 pm - OHE 122 and EEB 132 - room assignments to be announced
- One (two-sided) sheet of notes (8.5x11 or A4 paper), plus a simple calculator (not part of a
smart phone, iPod, iPad, etc.) are allowed - nothing else is allowed
- The actual exam will be 1.5 hours -- 10:05 am to 11:35 am
10-1 (15) Text problem 7.4 (Cauchy RV, characteristic function).
10-2 (15) Text problem 7.7 (Exponential RV, characteristic function). No inverse transforms
should be needed.
10-3 (20) A discrete random variable X is defined by
'
n
) A! 1 $ ,
) # &
n = 0, 2, 4, 6,...
P { X = n} = ( " 3 %
)
0 ,
otherwise
)
*
a). Find the value of A.
b). Find the characteristic function ? X (? ) .
2
c). Find E ( X ) and var(X) = ? X .
d). Find the conditional density P { X | X > 3} and E { X | X > 3} .
10-4 (15) X and Y are independent random variables. X is normal with mean m1 and variance
2
?12 , and Y is also normal with mean m2 and variance ? 2 . If Z is a new random variable
defined by Z = X + Y , use characteristic functions to show that Z is also normally
distributed, and compute its probability density, mean and variance in terms of m1, m2, ?12 ,
2
? 2 . (This is the same as problem 9-5 with a different method of solution).
10-5 (15) Text problem 7.30 (CLT).
1
10-6 (20) Joe Zilch has found that the hours spent working (W) and sleeping (S) in preparation
for an exam are random variables described by
$e ?s , 0 ? w ? 1 and 0 ? s ? ?
f S,W (s,w ) = %
elsewhere
&0,
It turns out that Joe?s grade (G) on the exam is helped more by his working than by his
sleeping, and that his grade is described by
G = S + 2W
a). Find and make a labeled sketch of the probability density function fG ( g) for all values
of G.
?
b). Find the conditional density of G given that Joe?s grade G was less than 1.
10-7 (15) The lifetime Xi of an iPod i (measured in years) is normally distributed with mean
E(Xi ) = 3 and variance ? 2 i= 0.25 . 100 iPods are selected independently and their total
X
lifetime Y is computed as
?
?
100
Y = ? Xi
i=1
Using the central limit theorem, find the approximate probability that Y is within 4 years of
its mean E(Y ) .
----------------------------?
Useful series expansions:
x2 x3 x 4
e x = 1+ x + + + ..............for all real x
2! 3! 4!
1
= 1+ x + x 2 + x 3 ................for x < 1
1? x
?
1
2
3
2 = 1+ 2x + 3x + 4x + .................for x < 1
(1? x)
x2 x 3 x 4
?ln(1? x) = x + + + .................for x < 1
2 3 4
?
Derivative of the quotient of two variables:
" u % vdu ? udv
d$ ' =
?
#v&
v2
?
2
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