Page 176 - Applied statistics and probability for engineers

154 Chapter 4/Continuous Random Variables and Probability Distributions

Mind-Expanding Exercises

4-227. The steps in this exercise lead to the probability den- 4-232. Determine the mean and variance of a beta random
sity function of an Erlang random variable Xwith parameters variable. Use the result that the probability density function

x e
λ and r, f x ( ) = λ r r−1 − x λ ( / r − )! , x > , and r = 1 , ,…. integrates to 1. That is,
2
0
1
(
(a) Use the Poisson distribution to express P X > x). Γ( ) Γ( ) 1 x α−1 ( − x ) for α > , > 0.
β
β
α
β−1
0
1
+
(b) Use the result from part (a) to determine the cumulative Γ(α β ) = ∫ 0
distribution function of X. 4-233. The two-parameter exponential distribution uses a dif-
γ
(c) Differentiate the cumulative distribution function in part (b) ferent range for the random variable X, namely, 0 ≤ ≤ x for
and simplify to obtain the probability density function of X. a constant γ (and this equals the usual exponential distribution
4-228. A bearing assembly contains 10 bearings. The bearing in the special case that γ = 0). The probability density func-
γ
tion for X is f x( ) = λ exp[−λ x ( − γ )] for 0 ≤ ≤ x and 0 < λ.
diameters are assumed to be independent and normally distrib-
uted with a mean of 1.5 millimeters and a standard deviation Determine the following in terms of the parameters λ and γ:
γ
1
(
of 0.025 millimeter. What is the probability that the maximum (a) Mean and variance of X. (b) P X < + / λ )
diameter bearing in the assembly exceeds 1.6 millimeters? 4-234. A process is said to be of six-sigma quality if the pro-
4-229. Let the random variable X denote a measurement cess mean is at least six standard deviations from the nearest
from a manufactured product. Suppose that the target value for speciication. Assume a normally distributed measurement.
the measurement is m. For example, X could denote a dimen- (a) If a process mean is centered between upper and lower
sional length, and the target might be 10 millimeters. The speciications at a distance of six standard deviations from
quality loss of the process producing the product is deined to each, what is the probability that a product does not meet
(
2
be the expected value of k X − m) , where k is a constant that speciications? Using the result that 0.000001 equals one
relates a deviation from target to a loss measured in dollars. part per million, express the answer in parts per million.
(a) Suppose that X is a continuous random variable with (b) Because it is dificult to maintain a process mean centered
E X ( ) = m and V X ( ) = σ . What is the quality loss of the between the speciications, the probability of a product
2
process? not meeting speciications is often calculated after assum-
(b) Suppose that X is a continuous random variable with ing that the process shifts. If the process mean positioned
2
E X ( ) = μ and V X ( ) = σ . What is the quality loss of the as in part (a) shifts upward by 1.5 standard deviations,
process? what is the probability that a product does not meet speci-
4-230. The lifetime of an electronic ampliier is modeled ications? Express the answer in parts per million.
as an exponential random variable. If 10% of the ampliiers (c) Rework part (a). Assume that the process mean is at a
have a mean of 20,000 hours and the remaining ampliiers distance of three standard deviations.
have a mean of 50,000 hours, what proportion of the ampli- (d) Rework part (b). Assume that the process mean is at
iers will fail before 60,000 hours? a distance of three standard deviations and then shifts
4-231. Lack of Memory Property. Show that for an expon- upward by 1.5 standard deviations.
(
ential random variable X, P X < 1 + t 2 u X > 1) = ( t (e) Compare the results in parts (b) and (d) and comment.
t
P X < 2).
t
Important Terms and Concepts
Beta random variable Exponential random Mean-function of a continuous Standard deviation-continuous
Chi-squared distribution variable random variable random variable
Continuity correction Gamma function Normal approximation to Standardizing
Continuous uniform Gamma random variable binomial and Poisson Standard normal random
distribution Gaussian distribution probabilities variable
Continuous random variable Lack of memory property- Normal random variable Variance-continuous random
Continuous uniform random continuous random Poisson process variable
variable variable Probability density function Weibull random variable
Cumulative distribution Lognormal random variable Probability distribution-
function Mean-continuous random continuous random
Erlang random variable variable variable