Concept explainers
The lifetime, in years, of a certain type of pump is a random variable with
- a. What is the probability that a pump lasts more than two years?
- b. What is the probability that a pump lasts between two and four years?
- c. Find the
mean lifetime. - d. Find the variance of the lifetimes.
- e. Find the cumulative distribution function of the lifetime.
- f. Find the median lifetime.
- g. Find the 60th percentile of the lifetimes.
a.
Find the probability that a pump lasts more than two years.
Answer to Problem 18E
The probability that a pump lasts more than two years is
Explanation of Solution
Given info:
The lifetime, in years, of a certain type of pump is a random variable with probability density function
Calculation:
Let X be a continuous random variable with probability density function
The probability that a pump lasts more than two years can be obtained as,
Thus, the probability that a pump lasts more than two years is
b.
Find the probability that a pump lasts between two and four years.
Answer to Problem 18E
The probability that a pump lasts between two and four years is
Explanation of Solution
Calculation:
Let X be a continuous random variable with probability density function
Let X be the lifetime, in years, of a certain type of pump.
The probability that a pump lasts between two and four years can be obtained as,
Thus, the probability that a pump lasts between two and four years is
c.
Find the mean life time.
Answer to Problem 18E
The mean lifetime is
Explanation of Solution
Calculation:
Mean:
Let X be a continuous random variable with probability density function
The mean lifetime can be obtained as,
Upper limit: if
Lower limit: if
Thus, the mean lifetime is
d.
Find the variance of the lifetimes.
Answer to Problem 18E
The variance of the lifetimes is
Explanation of Solution
Calculation:
Variance:
Let X be a continuous random variable with probability density function
The variance of the lifetimes can be obtained as,
Upper limit: if
Lower limit: if
Thus, the variance of the lifetimes is
e.
Find the cumulative distribution function of the lifetimes.
Answer to Problem 18E
The cumulative distribution function of the lifetimes is,
Explanation of Solution
Calculation:
Cumulative distribution function:
Let X be a continuous random variable with probability density function
For
For
Thus, the cumulative distribution function of the resistance is,
f.
Find the median of the lifetimes.
Answer to Problem 18E
The median of the lifetimes is 0.3784.
Explanation of Solution
Calculation:
Median:
The median is the half of the observations. Therefore,
The median of the lifetimes is,
Thus, the median of the lifetimes is 0.3784.
g.
Find the 60th percentile of the lifetimes.
Answer to Problem 18E
The 60th percentile of the lifetimes is 0.2745.
Explanation of Solution
Calculation:
Percentile:
The nth percentile can be written as
The 60th percentile of the lifetimes is,
Thus, the 60th percentile of the lifetimes is 0.2745.
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Chapter 2 Solutions
Statistics for Engineers and Scientists
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