Page 63 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 49

(
f x 0.1) = 0.1e – 0.1x .
;
a. What is the mean lifetime of the flat panel display?
b. What is the probability that the display fails within the first two
years?
c. Given that the display has been operating for one year, what is
the probability that it will fail within the next year?

2.7. The time to failure for a widget follows a Weibull distribution, with
⁄
ν = 0 , β = 12 , and α = 750 hours.
a. What is the mean time to failure of the widget?
b. What percentage of the widgets will fail by 2500 hours of oper-
ation? That is, what is the probability that a widget will fail
within 2500 hours?
2.8. Let’s say the probability of having a boy is 0.52. Using the Multipli-
cation Rule, find the probability that a family’s first and second chil-
dren are boys. What is the probability that the first child is a boy and
the second child is a girl?
2.9. Repeat Example 2.1 for n = 6 and p = 0.5. What is the shape of the
distribution?
(
2.10. Recall that in our piston ring example, PM A ) = 0.6 and
(
PM B ) = 0.4. From prior experience with the two manufacturers, we
know that 2% of the parts supplied by manufacturer A are likely to
fail and 6% of the parts supplied by manufacturer B are likely to fail.
(
(
Thus, PF M A ) = 0.02 and PF M B ) = 0.06. If we observe a piston
ring failure, what is the probability that it came from manufacturer A?
2.11. Using the functions fminbnd or fmin (available in the standard
MATLAB package), find the value for x where the maximum of the
,
N 31) probability density occurs. Note that you have to find the
(
minimum of fx()– to find the maximum of f x() using these functions.
Refer to the help files on these functions for more information on
how to use them.
2.12. Using normpdf or csnormp, find the value of the probability density
for N 01,( ) at ∞± . Use a small (large) value of x for ∞– (∞ ).
2.13. Verify Equation 2.38 using the MATLAB functions factorial and
gamma.
2.14. Find the height of the curve for a normal probability density function
,,
at x = µ , where σ = 0.512. What happens to the height of the
curve as σ gets larger? Does the height change for different values
µ
of ?
2.15. Write a function that calculates the Bayes’ posterior probability given
a vector of conditional probabilities and a vector of prior probabilities.

© 2002 by Chapman & Hall/CRC