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Unformatted text preview: ELEC210 Spring 2 011 Homework7Solutions
1. Let
X , X be the jointly Gauss
sian random variables with mean vecto r and covariance
matrix given by:
9/5
2/5 1
, 1 2/5
6/5 (a) Find the pdf of
in matrix notati on.
(b) Find the pdf of
using the quadr atic expression in the exponent.
(c) Find the marginal pdfs of X and X .
d
Solutions (20’)
(a) According to f  the matrix notation is f ex p 1/5
, det
9/1 0 3/5 1/5 1
, 1 and  √ ex p 1 (b)
3
5
Therefore, f
(c) According to f
Therefore f
2. Let X , … , X 1
√ √ 2
5 1 x
x 1 1
m 1 9
10 1 1 exp exp 1
1 1 exp 1 where
, √ ,f 1
, exp x 1 be the random variable s with the same mean μ and with cova riance function:
COV X , X where   2 if i j
σ if i j 1
ρσ
0
otherwise 1. Find the mean and var iance of Solutions (10’) ⋯ . 3. Suppose that 30% of voters are in fav or of certain legislation. A large numb er n of voters are
polled and a relative frequency estima te f
for the above proportion is obtained. Use the
following formula
P M μ ε 1 to determine how many voters should be polled in order that the probabilit is at least 0.90
ity
that f
differs from 0.30 by less th an 0.02.
Solutions (10’)
P M μ where ε 1 and thus P M
μ μ μ 0.21, μ ε
0.3, 0.1
0.02 n 4. A discretetime random process is de fined by X
s , for n
random from the interval 0,1 .
(a) Sketch some sample paths of the p rocess.
(b) Find the cdf of X .
(c) Find the joint cdf for X and X .
(d) Find the mean and autocovarianc e functions of X .
Solutions (20’) 5 250 0, where s is selected at 5. Let g t be the rectangular pulse sho wn in the figure. The random proces s X t is defined
s
as
Xt
Ag t
Where A assumes the values 1 wit h equal probability. (a)
(b)
(c)
(d) Find the pmf of X t .
Find m t .
Find the joint pmf of X t and X t
Find C t, t d , d 0. Solutions (20’) d. For t ∉ 0,1 , t d ∈ 0,1 , P x t 0, x t d 1 6. A student uses pens whose lifetime is an exponent random variable with m ean 2 weeks. Use
s
the central limit theorem to determin e the minimum number of pens he s hould buy at the
beginning of a 15week semester, so that with probability 0.99 he does no t run out of pens
during the semester.
Hint: 2.3263 0.99 Solutions (20’)
Denote
as the lifetime of the ith pe n, and
⋯
pens.
∙2 2 By independence between the lifetime of different pens,
VAR
Assuming VAR
4
∙2
is approximately Gauss ian, and by applying CLT: 15
From hint:
17.12 √ √ √ √ 0.99 2.3263 3.28 and this student sh ould buy at least 18 pens. as the total lifetime of the ...
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This note was uploaded on 10/16/2011 for the course ELEC 308,315,10 taught by Professor Prof.shenghuisong during the Spring '11 term at CUHK.
 Spring '11
 Prof.ShenghuiSong

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