Concept explainers
A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with
a. Find P(X1 > 100).
b. Find P(X1 > 100 and X2 > 100 and … and X5 > 100).
c. Explain why the
d. Find P(T ≤ 100).
e. Let t be any positive number. Find P(T ≤ t), which is the cumulative distribution
f. Does T have an exponential distribution?
g. Find the mean of T.
h. If there were n lightbulbs, and the lifetime of each was exponentially distributed with parameter λ, what would be the distribution of T?
a.
Find the value of
Answer to Problem 11E
The value of
Explanation of Solution
Given info:
Total number of lightbulb is 5. The lifetime of each bulb is exponentially distributed with mean 200 hours. The random variable T is defined as the time of the first bulb replacement. The random variables
Calculation:
The random variables
Exponential distribution:
The probability density function of the exponential distribution with parameter
Mean of anExponentialrandom variable:
The random variable X has Exponential distribution with parameter
Substitute 200 for
Thus, the parameter is 0.005.
The cumulative distribution function of the exponential distribution with parameter
Substitute 0.005 for
Thus, the value of
b.
Find the value of
Answer to Problem 11E
The value of
Explanation of Solution
Calculation:
Here, the random variables
Then the joint probability density function is the product of the marginal, each of which is an
That is,
Substitute 0.005 for
Thus, the value of
c.
Explain the reason behind the event
Explanation of Solution
The random variable T is defines as the time of the first bulb replacement.
The random variables
Thus, the time of the first replacement will be greater than 100 hours if and only if each of the bulb lasts longer than 100 hours.
Hence, the event
d.
Find the value of
Answer to Problem 11E
The value of
Explanation of Solution
Calculation:
The random variable T is defined as the time of the first bulb replacement.
From part (b), the value of
From, part(c), it is clear that the event
Then,
Substitute
Thus, the value of
e.
Find the cumulative distribution function of T, that is
Answer to Problem 11E
The cumulative distribution function of T is,
Explanation of Solution
Calculation:
Here the random variables
Then the joint probability density function is the product of the marginal, each of which is an
That is,
From, part(c), it is clear that the event
Then,
Substitute 0.005 for
Thus, the value of
f.
Check whether T has an exponential distribution or not.
Answer to Problem 11E
The random variable T follows exponential distribution with parameter
Explanation of Solution
If the random variable X follows exponential distribution with parameter
From (e), the cumulative distribution function of T is,
Thus, the random variable T follows exponential distribution with parameter
g.
Find the mean of T.
Answer to Problem 11E
The mean of T is40 hours.
Explanation of Solution
Calculation:
Mean of anExponentialrandom variable:
The random variable T has Exponential distribution with parameter
Substitute 0.025 for
Thus, the mean of T is40 hours.
h.
Find the distribution of T, if there were n lightbulbs and the lifetime of each was exponentially distributed with parameter
Answer to Problem 11E
The distribution of T is exponential with parameter
Explanation of Solution
Calculation:
The random variable T is defined as the time of the first bulb replacement.
The random variables
Here the random variables
Then the joint probability density function is the product of the marginal, each of which is an
From, part(e),
Thus, the cumulative distribution function of T is,
Thus, the random variable T follows exponential distribution with parameter
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Chapter 4 Solutions
Statistics for Engineers and Scientists
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageGlencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillAlgebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal Littell