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Chapter 5: Q22E (page 346)

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\(\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = 1 - F\left( x \right)\)

Short Answer

Expert verified

\(\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = 1 - F\left( x \right)\)

Step by step solution

01

Given information

X be the random variable with \(f\left( x \right)\) as the pdf and \(F\left( x \right)\)as the cdf.

02

Verifying \(\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = 1 - F\left( x \right)\)

Let us consider the integral

\(\int\limits_0^x {h\left( t \right)dt = \int\limits_0^x {\frac{{f\left( t \right)}}{{1 - F\left( t \right)}}dt} } \)

F is the cdf of X, hence

\(\begin{array}{l}F\left( x \right) = \int\limits_0^x {f\left( t \right)dt} \\\frac{d}{{dt}}\left( {1 - F\left( t \right)} \right) = - f\left( t \right)\end{array}\)

Hence by using the integration by parts

\(\begin{aligned}{}\int\limits_0^x {h\left( t \right)dt} &= - \left[ {\log \left( {1 - F\left( t \right)} \right)} \right]_0^x\\ &= - \left[ {\log \left( {1 - F\left( x \right)} \right) - \log \left( {1 - F\left( 0 \right)} \right)} \right]\\ &= - \log \left( {1 - F\left( x \right)} \right)\end{aligned}\)

Substitute in the left-hand side of equation (1).

\(\begin{array}{l}\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = \exp \left[ {\left( { - \log \left( {1 - f\left( x \right)} \right)} \right)} \right]\\\exp \left[ { - \int\limits_0^x {h\left( t \right)dt} } \right] = 1 - F\left( x \right)\end{array}\)

Hence, we have proved the required equality.

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Most popular questions from this chapter

Suppose that a random variable\(X\)has the normal distribution with mean\(\mu \)and variance\({\sigma ^2}\). Determine the value of\(E\left[ {{{\left( {X - \mu } \right)}^{2n}}} \right]\)for\(n = 1,2....\)

If the temperature in degrees Fahrenheit at a certain location is normally distributed with a mean of 68 degrees and a standard deviation of 4 degrees, what is the distribution of the temperature in degrees Celsius at the same location?

Suppose that two players A and B are trying to throw a basketball through a hoop. The probability that player A will succeed on any given throw is p, and he throws until he has succeeded r times. The probability that player B will succeed on any given throw is mp, where m is a given integer (m = 2, 3, . . .) such that mp < 1, and she throws until she has succeeded mr times.

a. For which player is the expected number of throws smaller?

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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 five components are functioning simultaneously, that the lifetimes of the components are i.i.d., and that each life has the exponential distribution with parameter\(\beta \). Let\({T_1}\)denote the time from the beginning of the process until one of the components fails, and let\({T_5}\)denote the total time until all five components have failed. Evaluate\(Cov\left( {{T_{1,}}{T_5}} \right)\)
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