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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