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ECE 341 - Probability and Random Processes for Engineers
Take-Home Final Examination
Due Friday July 25, 2003 in room 1020 SEO by 1:00 PM
14 problems, not all equally weighted.
Individual work is required; getting help or helping others is considered to be academically dishonest.
Please be neat; show all work. Problems with no work shown receive zero credit even if the answer is correct.
IMPORTANT: Please be aware that academic dishonesty will result in grade “E” for the course, and possibly dismissal from the University. Looking at another exam paper, showing someone your exam paper, or otherwise communicating exam problem information to/from another student is considered to be academically dishonest. Should you have any questions about this policy or about any of the exam questions, please ask me for clarification. -Vladimir Goncharoff
In the Venn Diagrams showing sample space S below, draw and label shapes representing events A and B (having areas proportional to probability) for the specified conditions.
a) A and B are independent, P[ A ] = 1/2, and P[ A C^ B ] = 7/8:
b) A and B form an event space and P[ A ] = 1/8:
c) A and B are collectively exhaustive, P[ A B ] = 1/2, and P[ B ] = 3/4:
d) A and B are mutually exclusive, P[ A ] = 1/2, and P[ B C ] = 3/4:
S
S
S
S
Binomial random variable X , having parameters n = 10 and p = 1 / 3 , is sampled
five times to yield outcomes { x 1 , x (^) 2 , x 3 , x (^) 4 , x 5 } in that order.
a) What is the probability that x 1 = x 2 = x 3 = x 4 = x 5 = 0?
b) What is the probability that x 1 = 0, x 2 = 1, x 3 = 2, x 4 = 3, x 5 = 4?
c) What is the probability that the samples are {0, 1, 2, 3, 4}, in any order?
d) What is the probability that none of the five samples is equal to 10?
Problem 4.
Geometric random variable Y , having parameter p = 1 / 3 , is sampled five times.
a) What is the probability that none of the five outcomes is less than 5?
b) What is the probability that none of the five outcomes is greater than 5?
A particular operation has six components. Components {1, 2, 3} each have success probability p A, and components {4, 5, 6} each have success probability p B. The operation is successful if {components 1, 2 and 3 all work} and {at least one of the remaining three components works}.
a) Write an expression of the probability that the operation is successful in terms of p A and p B :
b) When price we pay for each of components {1, 2, 3} is ($5.00 ÷ (1 − p A)), and the price we pay for each of components {4, 5, 6} is ($1.00 ÷ (1 − p B )), what is the best choice of values for p A and p B when have $100.00 to spend and wish to have the highest possible probability that the operation is successful?
p A =
p B =
Continuous random variable X , uniformly distributed over range −5 to +5 volts, is fed into an amplitude limiter to give random variable Y. The relationship between X and Y is:
X
X X
X
Y
a) Plot f Y ( y ), using impulse functions wherever necessary:
b) What is Cov [ X , Y ]?
( Hint: consider cases X <− 3 , − 3 ≤ X ≤ 3 , and X > 3 separately and get the final result as a weighted average of the covariances corresponding to each case )
Continuous uniform random variable X has range −3 to +2.
a) What is μ (^) X?
b) What is E [ X^2 ]?
c) What is P [ X > 0 ]?
d) Write one line of Matlab code to generate 1000 samples of X :
Problem 9.
Continuous Gaussian random variable X has μ (^) X =− 1 and σ (^) X = 10.
a) What is Cov [ X ,− 6 X ]?
b) What is P [ X > 0 ]?
c) Write one line of Matlab code to generate 1000 samples of X :
A sequence of pulses is used to transmit bits of information over a communication
channel having additive Gaussian noise interference. The pulse amplitudes are − A
and + A volts, equally likely to occur and determined as independent samples of a
discrete random variable. The received noisy pulses are each sampled once and
then rounded to their nearest original amplitude level in an attempt to recover the
transmitted bit stream. We are told that on the average 1 out of every 1000 bits is
incorrectly received when A = 1.
a) What value of A will give channel SNR = 25 dB?
b) Assume we transmit 1 million pulses per second. What value of A will result in
one bit error per year, on the average?
Problem 13.
We are given n samples of random variable X : { x 1 (^) , x 2 , , xn }. Mean value μ (^) X is
not known. Based on this information write a formula that will most accurately
estimate the variance of X :
Var [ X ] ≅
Problem 14. (Short answers, please)
a) When do we label a random process "wide-sense stationary"?
b) When do we label a random process "strictly stationary"?
c) When do we label a random process "ergodic"?
d) What is the advantage of knowing that a given random process is ergodic?
e) How may the distribution of signal power vs. frequency be found from the statistics of a wide-sense stationary random process?
