Page 62 - Computational Statistics Handbook with MATLAB

48 Computational Statistics Handbook with MATLAB

Exercises

2.1. Write a function using MATLAB’s functions for numerical integration
such as quad or quadl (MATLAB 6) that will find PX ≤( x) when
λ
the random variable is exponentially distributed with parameter .
See help for information on how to use these functions.
2.2. Verify that the exponential probability density function with param-
eter λ integrates to 1. Use the MATLAB functions quad or quadl
(MATLAB 6). See help for information on how to use these functions.
2.3. Radar and missile detection systems warn of enemy attacks. Suppose
that a radar detection system has a probability 0.95 of detecting a
missile attack.
a. What is the probability that one detection system will detect an
attack? What distribution did you use?
b. Suppose three detection systems are located together in the same
area and the operation of each system is independent of the others.
What is the probability that at least one of the systems will detect
the attack? What distribution did you use in this case?
2.4. When a random variable is equally likely to be either positive or
negative, then the Laplacian or the double exponential distribution
can be used to model it. The Laplacian probability density function
for λ > 0 is given by

1 – λ x
-
f x() = --λe ; – ∞ < x < ∞ .
2
a. Derive the cumulative distribution function for the Laplacian.
b. Write a MATLAB function that will evaluate the Laplacian proba-
bility density function for given values in the domain.
c. Write a MATLAB function that will evaluate the Laplacian cumu-
lative distribution function.
d. Plot the probability density function when λ = . 1

2.5. Suppose X follows the exponential distribution with parameter λ .
Show that for s ≥ 0 and t ≥ 0 ,

(
P X > s + t X > s) = PX > . t)
(
2.6. The lifetime in years of a flat panel display is a random variable with
the exponential probability density function given by

© 2002 by Chapman & Hall/CRC