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(ELEC210)[2008](f)quiz~419^_10291.pdf
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Your Name: _____________________ Student ID: _____________________ ELEC210 C Probability and Random Processes in Engineering Quiz 2
November 18, 2008 Duration: 45 minutes There are 10 multiple-choice questions. You receive 3 points for each correct answer, 1 point for abstaining, and 0 point for a wrong answer.
ANSWER SECTION (Write down a, b, c, d as an answer or leave blank to abstain)
(1)
______ (2) ______ (3) ______ (4) ______ (5) ______

(6)
______ (7) Below__ (8) ______ (9) Below _ (10) ______

Written Answer:

(7)

(9)


1.
X is a random variable with mean equals to 3 and variance equals to 2. The mean and variance of Y = 2X- 3 are, respectively a) 3, 6 b) 6, 1 c) 3, 8 d) 6, 6

2.
The random variables Xand Y have the joint pmf as shown below: X


0 1 2
0
Y 1
2
1/20 2/20 2/20
2/20 4/20 4/20
1/20 2/20 2/20

COR(X,Y), the correlation between Xand Yis
a) 0.5 b) 1
b) 1.2 d) 2

3. Two random variables are uncorrelated if their covariance, [( .[ ])( .[ ])]Y
EX EX YE, is zero. For the joint pmf in Question 2, Xand Yare: a) Dependent and uncorrelated b) Dependent and correlated c) Independent and uncorrelated d) Independent and correlated
JJG
4. For a random vector X, we assume that the covariance matrix is K. X1 1 -0.3 0.5 .

.. .
JJG
.. ..
X=X2 K=- 0.3 2 0.3
.. .. X 0.5 0.3 1 .
...


..3 ..
XX X
The variance of the random variable 1 +2 +3 is:
a) 4 b) 5
c) 1.2 d) 1.8

.1.
JG

JJG ..
m =2
5. Let the mean vector of X in Question 4 be ..
.3.
..
Assume that X1 and X2 are jointly Gaussian. Which of the followings represents the plot of the constant value contour of the joint pdf of X1 and X2


6. The characteristic function of a random variable X is EejX ]
[ . The characteristic function of the Bernoulli random variable with probability of success of p and the characteristic function of the binomial random variable that results from repeating the Bernoulli trial n times are, respectively,
j jnj jn
a) 1.+p pe , (1 ppe ) b) pe ,
.+pe
j j j j
c) 1.+p pe , n(1.+pe ) d) pe ,
p npe
7. The random vector (X, Y) has a joint pdf that is non-zero in the region xy
>>0 (such a triangular region is known as a cone). .ax /2 .by /2fXY (, ) =caxe 22 xy 0, a >0,b >0 ; zero otherwise.


x y bye >>
where c is a normalizing constant
Are X and Y independent? Briefly explain.

8. For the joint density in Question 7, the marginal pdf of yis:
2 .+ 2/2
.+(aby )/2 (aby )a) Y b) Y
() =cbyfy=ye

fy e()
.by2/2 .by2/2c) Y d) Y
() =befy=ye

fy y()
9.
Given the joint density in Question 7, specify the conditional pdf of Xgiven Y, fXY|(|xy).

Show how you come up with the answer. Be very careful about the range of Xfor which the conditional pdf is non-zero.

10.
Assume that the inter-arrival times of photon at a detector are exponentially distributed with a mean of l nanosecond. Recall