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Chapter 5: Q20E (page 316)
LetXhave the lognormal distribution with parameters 3 and 1.44. Find the probability thatX鈮�6.05.
颅\[P\left( {X \le 6.05} \right) = 0.1587\]
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Suppose that X is a random variable having a continuous distribution with p.d.f.\(f\left( x \right)\)and c.d.f.\(F\left( x \right)\)and for which\({\rm P}\left( {X > 0} \right) = 1\)Let the failure rate\(h\left( x \right)\) be as defined in Exercise 18 of Sec. 5.7. Show that\(\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = 1 - F\left( x \right)\)
Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}\) are independent and that \({{\bf{X}}_{\bf{i}}}\) has the Poisson distribution with mean \({{\bf{\lambda }}_{\bf{i}}}\left( {{\bf{i = 1, \ldots ,k}}} \right)\). Show that for each fixed positive integer n, the conditional distribution of the random Vector \({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}} \right)\), given that \(\sum\limits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{X}}_{\bf{i}}}{\bf{ = n}}} \) it is the multinomial distribution with parameters n and
\(\begin{array}{l}{\bf{p = }}\left( {{{\bf{p}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{p}}_{\bf{k}}}} \right){\bf{,}}\,{\bf{where}}\\{{\bf{p}}_{\bf{i}}}{\bf{ = }}\frac{{{{\bf{\lambda }}_{\bf{i}}}}}{{\sum\limits_{{\bf{j = 1}}}^{\bf{k}} {{{\bf{\lambda }}_{\bf{j}}}} }}\,{\bf{for}}\,{\bf{i = 1, \ldots ,k}}{\bf{.}}\end{array}\)
Suppose that a certain system contains three components that function independently of each other and are connected in series, as defined in Exercise 5 of Sec. 3.7, so that the system fails as soon as one of the components fails. Suppose that the length of life of the first component, measured in hours, has the exponential distribution with parameter\(\beta = 0.001\), the length of life of the second component has the exponential distribution with parameter\(\beta = 0.003\), and the length of life of the third component has the exponential distribution with parameter\(\beta = 0.006\).Determine the probability that the system will not fail before 100 hours.
It is said that a random variable has the Weibull distribution with parameters a and b (a > 0 and b > 0) if X has a continuous distribution for which the p.d.f. f (x|a, b) is as follows:
\({\bf{f}}\left( {{\bf{x|a,b}}} \right){\bf{ = }}\frac{{\bf{b}}}{{{{\bf{a}}^{\bf{b}}}}}{{\bf{x}}^{{\bf{b - 1}}}}{{\bf{e}}^{{\bf{ - }}{{\left( {\frac{{\bf{x}}}{{\bf{a}}}} \right)}^{\bf{b}}}}}\,{\bf{,x > 0}}\)
Show that if X has this Weibull distribution, then the random variable \({{\bf{X}}^{\bf{b}}}\) has the exponential distribution with parameter \({\bf{\beta = }}{{\bf{a}}^{{\bf{ - b}}}}\)
If the m.g.f. of a random variable X is\(\psi \left( t \right) = {e^{{t^2}}}\,for - \infty < t < \infty \)What is the distribution of X?
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