University of Illinois
Fall 2009
ECE 313:
Problem Set 10
Exponential and Gaussian Random Variables; Poisson Processes
Due:
Wednesday November 4 at 4 p.m.
Reading:
Ross, Chapter 5; Powerpoint Lecture Slides, Sets 24-27
Qfunction
: Note available on the COMPASS web page
Noncredit Exercises:
Chapter 5:
Problems 7, 11, 12 15-24, 32, 33;
Theoretical Exercises 2, 5, 8, 9; Self-Test Problems 8-11
1.
[Reliability function of a triple-modular-redundancy (TMR) system]
Consider a triple modular redundancy (TMR) system (cf. Lecture 19 of the Powerpoint slides) with a
perfect majority gate. The three modules have lifetimes denoted by
X
1
,
X
2
,
X
3
and fail independently
of each other, that is, as in Problem 5 of Problem Set 9, the events
{
X
1
> t
1
}
,
{
X
2
> t
2
}
, and
{
X
3
> t
3
}
are independent events for all choices of
t
1
,
t
2
, and
t
3
. The modules are identical and hence we assume
that they have identical reliability functions:
P
{
X
i
> t
}
=
R
(
t
) for
i
= 1
,
2
,
3. Let
Y
denote the
lifetime of the TMR system.
(a) Express the event
{
Y
> t
}
in terms of unions, intersections and complements of the events
{
X
1
> t
}
,
{
X
2
> t
}
,
{
X
3
> t
}
and use this result to express the reliability function
R
Y
(
t
) of the