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The lifetime of a product produced at Hewlett-Packard is modeled using an exponential distribution with a mean of 500 thousand hours. During the normal (useful) life, the product time till failure is uniformly distributed 0ver the range of 100 thousand to 1 million hours.

Let X denotes the length of time till the failure of a product.

The probability density function of a random variable X is:

$f\left(x\right)=\frac{1}{500}{e}^{\frac{-x}{500}};x>0$.

Let Y denotes the product鈥檚 time till failure during its normal life

The probability density function of Y is:

$\begin{array}{rcl}f\left(y\right)& =& \frac{1}{1000-100};100猢�x猢�1000\\ & =& \frac{1}{900};100猢�x猢�1000\end{array}$.

a.

At the end of the product鈥檚 lifetime, the probability that the product fails before 700 thousand hours is obtained as:

$\begin{array}{rcl}P\left(X猢�700\right)& =& 1-P\left(X>700\right)\\ & =& 1-e^{\frac{-700}{500}}\\ & =& 1-e^{-1.4}\\ & =& 1-0.2466\\ & =& 0.7534\end{array}$

For the exponential distribution:$\mathit{P}\left(X猢�a\right)\mathbf{=}{\mathit{e}}^{\frac{\mathbf{-}\mathbf{a}}{\mathbf{胃}}}$.

Therefore, the required probability is 0.7534.

b.

During its normal life, the probability that the product fails before 700 thousand hours is obtained as:

$\begin{array}{rcl}P\left(Y猢�700\right)& =& \frac{700-100}{900}\\ & =& \frac{600}{900}\\ & =& 0.6667\end{array}$

For the uniform distribution: $\mathit{P}\left(\mathit{a}<\mathit{x}<\mathit{b}\right)\mathbf{=}\frac{\mathbf{b}\mathbf{-}\mathbf{a}}{\mathbf{d}\mathbf{-}\mathbf{c}}\mathbf{,}\mathit{c}\mathbf{猢�}\mathit{a}\mathbf{<}\mathit{b}\mathbf{猢�}\mathit{d}$.

Therefore, the required probability is 0.6667.

c.

The probability of the product failing before 830 thousand hours is:

$\begin{array}{rcl}P\left(X猢�830\right)& =& 1-P\left(X>830\right)\\ & =& 1-e^{\frac{-830}{500}}\\ & =& 1-e^{-1.6}\\ & =& 1-0.2019\\ & =& 0.7981\end{array}$

Also,

$\begin{array}{rcl}P\left(Y猢�830\right)& =& \frac{830-100}{900}\\ & =& \frac{730}{900}\\ & =& 0.8111\\ & & \end{array}$

Therefore, the probabilities are approximately the same for the normal (useful) life distribution and the wear-out distribution.