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ELEC 2600 Probability and Random Processes in Engineering
2012 Spring Semester - Midterm Exam Paper with Marking Scheme March 16, 2012
SECTION I: Multiple Choice [20 Marks]

1. [4] Which of the following statements is wrong? ( B )
A. The sub-experiments in a sequential experiment can be either dependent or independent;
B. A uniform random variable has flat PDF, so its variance will be 0;
C.

D.

The Binomial distribution describes the number of successes in several independent trails;
and .
Poisson distribution is an approximation to the Binomial distribution with large.

small
.


2.

[4] We perform n independent sequential experiments. In each experiment, therere three
the Whats..
..
.1...
. .
, and .
.....,.
...
.. B, and C. Also, outcomes A, ,.
.. times ( occurs a times and B occurs b.
experiments, nprobability that, among the
..
.
.

.!)?
.!.!
( D )

.

.
.;
A.

.

...! .!
...!
.

.;
.! .!
....
.
.
!
.
B.

.

.

.

.1.
.

.
.

.
.....
..!......!.
.
.
.!.!
.
.
..
!.!.!.!
..

C.

;

.1.....
..!.....
.
.....
.
D.

.

[4] We want to measure the distance from the redbird to the seafront cafeteria. For each
0..follows Gaussian distribution with .
error
0.1|.
|. What is the probability that .9

.
.. .
,....
(Hint:

3.

measurement, the
.
and variance
.
mean
?

( A)

.. ..
...
.
.

). .

.
.

......
......
.
.
.
.
A. 2.....;
.
2 2
B.
..
...;
.
1.2
..
...
C.

;

D. 1.2......
.
1

4. [4] Suppose we roll a fair dice (each digit occurs with equal probability) and we want to get a six. Whats the probability that it will take 4 more trials until we finally get a six for the first time, given that we have already thrown it for 2 times and get two non-six digits? ( C )
A. .......;
B. .......;
C. .......;
D. ........
,.0.4...
.... .
[4] In an experiment, one of the three events follows exponential distribution given .
. Random variable.0.5 ...
. .
, and .0.1 ...
,.5.76 ..VAR,.3.24 ..|.
.VARhappens.).1,2,3 .
... Whatis.0.36 .|.

..
.
2.052;
.
,
.
and
.
occurs. Also

5.

..

VAR
.
.

.|.

.
(
.
.
Besides,
.
and
?

( B )

A.

B. 1.26;
C. 2.1;
D. 1.10.


SECTION II: Problems [80 Marks]

1.

[20 Marks]
find the following probability. ,.0.6... .
,.0.7 ..
..


..0.9...
.. .
(a)If ;....
..[4] Compute ..
..
..[4] Compute ....
..[4] Compute ..
..
..[4] Compute (iv)
| .
, compute.0.56 .|. .
(b)[4] If
...
..
. .
,.0.7 ..
....
.......

.3.
.;.
.
(i)
(ii);
.
(iii);
.
.
.

.

.

.

.


Solution:
.
(a)Because ..

.
and .0.6 ...
.....
...
.
..0.9,.3...0.7.0.6.0.9.0.4.1.
.
(i)
;

.
....
....
.
.
.

..

.

...

..0.6.0.4.0.2.1..
.
(ii)
;

.
....
....
.
.

...

.
.

...

.3...0.7.0.