University Problem Set #10 ECE 313
of Illinois Page 1 of 2 Spring 2002
Assigned: Wednesday, March 27, 2002
Due: Wednesday, April 3, 2002
Reading: Ross, Chapter 5, Chapter 9.1
Noncredit Exercises: Ross, Chapter 5: Problems 11, 13, 15-38; Theoretical Exercises 12-16,
21, 26, 28-30
Problems:
1. Let X denote a unit Gaussian random variable with pdf φ(u) and CDF Φ(u).
(a) What is the derivative of exp(–u2/2) with respect to u? Use this result to find E[|X|].
Now, let Q(x) = ∫
x
∞
φ(u) du = ⌡
⌠
x
∞
( 2π)–1exp
– u2
2 du = 1 – Φ(x).
(b) Some tables list the values of Q(x) (instead of Φ(x)) for large values of x. Why might the
tabulator have chosen to specify Q(x) instead of Φ(x)? Explain briefly.
The rest of this problem leads you through a derivation of the bounds (1/x–1/x3)φ(x) < Q(x) < φ(x)/x
mentioned in class and it also asks you to prove at another simpler bound on Q(x).
(c) Write the integrand for Q(x) as ( 2π)–1u–1(u•exp(–u2/2)) and integrate by parts to deduce
that Q(x) < x–1•φ(x) for x > 0.. Repeat the trick of re–writing and integrating by parts to
show that x–1•φ(x) – x–3•φ(x) < Q(x). Are these bounds useful as x → 0? Why or why
not? What is the asymptotic value of the ratio of the bounds as x → ∞?
(d) A useful bound when x is small is Q(x) ≤ (1/2)exp(–x2/2) for x ≥ 0 in which equality holds
only at x = 0. Derive this bound by first showing that t2 – x2 > (t – x)2 for t > x > 0 and
then applying this result to exp(x2/2)Q(x) = ⌡
⌠
x
∞
( 2π)–1exp
–
t2 – x2
2 dt
(e) For what values of x is this smaller than the upper bound of Eq.(4.4)?
2. Do either part (a) or part (b). Then do parts (c)–(e).
(a) Attach to your homework a photocopy of your calculator’s manual page(s) that explains
which formula your calculator uses to compute Q(x). Reading the page might help too!
Note: I do not want to know which buttons you have to press in order to find Q(x); I
want to know what formula your calculator uses internally to find Q(x).
The xerographically–challenged are permitted to just copy the relevant formulas to their
homework. NEXT: press the appropriate buttons to find Q(5).
If your calculator cannot compute Q(x), or if the manual does not state what formula is used to calculate
Q(x) but just tells you which buttons to press, or if you have lost the manual, do part (b) instead.
(b) Read Chapter 26.2 of Abramowitz and Stegun (reference book (not a reserve book) in
Grainger Engineering Library), and use Equation 26.2.17 to calculate Q(5).
This formula is also given on Slide 30 of Lecture 26 in the Powerpoint slides.
(c) The number found in part (a) or (b) is just an approximation to the value of Q(5). Use the
maximum error specification to find the range in which the actual value of Q(5) must
necessarily lie. What is the maximum relative error in the approximation to Q(5) that you
found in part (a) or (b)? Note: the relative error is defined as |true value–computed value|
true value
expressed as a percentage.
(d) On p. 972, Abramowitz and Stegun give the value of –log10Q(5) to be 6.54265…. Blindly
trust your calculator to do the exponentiation correctly and find the actual relative error in
the approximation to Q(5) that you found in part (a) or (b). What would the actual relative
