a.
Obtain
a.
Answer to Problem 113SE
The expression for
The expression for
Explanation of Solution
Given info:
The fusion of two exponential distributions for the modeling behavior is termed as “criticality level of the situation” X by the authors.
The probability density
Calculation:
The
The variance of the exponential distribution is:
The expected mean is obtained as given below:
Thus, the expected mean for mixed exponential distributions is:
The variance of a continuous random variable is given as
Here,
Thus, the variance is:
Thus, the
b.
Find the cumulative distribution of X.
b.
Answer to Problem 113SE
The cumulative distribution function of X is:
Explanation of Solution
Denote the cumulative distribution function (cdf) of X as
For
Since X takes the positive values from the interval
For
Hence, the cumulative distribution function of X is:
c.
Find the value of
c.
Answer to Problem 113SE
The value of
Explanation of Solution
Calculation:
The value of
Substitute
Thus, the value of
d.
Find the probability that the X is within one standard deviation from its mean value.
d.
Answer to Problem 113SE
The probability that the X is within one standard deviation from its mean value is 0.879.
Explanation of Solution
Calculation:
From part (a), the expected mean for two exponential distributions is:
Substitute
The standard deviation of X is given by:
The probability that the X is within one standard deviation from its mean value is obtained as shown below:
Here, the random variable X do not takes negative value.
Thus, the probability that the X is within one standard deviation from its mean value is 0.879.
e.
Find the coefficient of variation for an exponential random variable.
Explain about the value of coefficient of variation for a hyper exponential distribution.
e.
Answer to Problem 113SE
The coefficient of variation for an exponential random variable is 1.
Explanation of Solution
Calculation:
It is given that the Coefficient of variation for a random variable X is:
From part (a),
On simplifying,
The coefficient of variation for random variable X in hyper exponential distribution is:
Denote
Then,
The algebraic expression shows that the value of
Thus, the value of coefficient of variation is greater than 1.
f.
Find the coefficient of variation for an Erlang distribution.
f.
Answer to Problem 113SE
The coefficient of variation for an Erlang distribution is
Explanation of Solution
Calculation:
The expected mean for the Erlang distribution is:
The standard deviation for the Erlang distribution is:
The coefficient of variation for an Erlang distribution is given by:
Thus, the coefficient of variation for the an Erlang distribution is
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Chapter 4 Solutions
Probability and Statistics for Engineering and the Sciences
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