ECE 3800: EXAM #2
1. [60 pts])
Consider a joint density function defined for random variable X and Y as follows:
4  2  x   y,
f XY  x, y   
0,
1  x  2, 0  y  1
otherwise
a.(10) Find the mean of X and Y:
b.(10) Find the 2nd Moment of X and Y
c.(10) Find the Variance of X and Y
d.(10) Find the correlation E XY 
e.(10) Find the Correlation Coefficient
f.i.(5) Are X and Y independent? Why?
f.ii.(5) Are X and Y Uncorrelated? Why?
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#2
Spring
2. [45 pts])
Consider the random process X t   2  A  cos2    t    , where A and are
independent random variables. Assume that A is a zero mean random variable with a variance of
1 and that  is a uniformly distributed random variable on   ,   .
a.(5)
Find the probabilistic mean EX t  based on A and 
b.(5)
Find the time average mean xt 

c.(10) Find the probabilistic 2nd moment E X t 
d.(10) Find the time average 2nd moment xt 
2

2
e.i.(5)
Is the random process stationary? Why or why not?
e.ii..(5)
Is the random process ergodic?
e.iii.(5)
Is the random process deterministic?
This material has not yet been covered in class.
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#2
Spring
3. [40 pts])
The resistance of precision resistors manufactured by a certain company is
claimed to have a mean value of resistance of 300 ohms.
a.(20) In order to justify this claim for “Order ZYX” a sample of 25 resistors is taken. For this
sample set, the mean is found to be 298.8 ohms with a statistically measured standard deviation
of 2.5 ohms. Is this claim justified at the 98% or 95% level? (Derive and define the appropriate
values to show true or false for each of the levels.)
b.(20) In order to justify this claim for “Order US Gov.” a sample of 225 resistors is taken. For
this sample set, the mean is found to be 299.8 ohms with a true probabilistically defined standard
deviation of 2 ohms. Is this claim justified at the 90% or 80% level? (Derive and define the
appropriate values to show true or false for each of the levels.)
ECE 3800, Probabilistic Methods of Signal and System Analysis
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ECE 3800 Exam#2
Spring
4. [35 pts])
Suppose X and Y are two resistors with tolerances that form independent random
variables. The X resistor is 100 ohms +/- 5% were the tolerance has a uniform probability
density (X=x + x). The Y resistor is 50 ohms +/-5%, again, where the tolerance is uniformly
distributed (Y=y + y). Note that the random variables are independent.
 1 , 5  x  5
 1 ,
 2 . 5   y  2 .5
and f y y    5
f x x    10
0,
0,
otherwise
otherwise
The resistors are placed in series so that, Z = X + Y or (z + z) = (x + x) + (y + y)
a.(15) Find the density function f z z  that forms the tolerance of the series resistance.
b.(5)
Find the mean of Z, that is (z + z).
c.(10) Find the variance of Z.
d.(5)
Is the variance of the series resistor greater than or less than that of a single “Z mean”
resistor with a +/- 5% tolerance, again, assuming a uniform probability density?
ECE 3800, Probabilistic Methods of Signal and System Analysis
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