91Ó°ÊÓ

When without bolt on-trace elements, the pipe wall temperature is uniform from $260°F to 290°F$

Let x be a random variable that follows uniform distribution.

The p.d.f is given below

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{290-260}; 260â\copyright ½xâ\copyright ½290\\ & =& \frac{1}{30}; 260â\copyright ½xâ\copyright ½290\end{array}$

When several bolt-on trace element are attached to the piping, the wall temperature has a uniform distribution ranging from $278°F to 285°F$.

The p.d.f is given by

$\begin{array}{rcl}f\left(x\right)& =& \frac{1}{285-278}; 278â\copyright ½xâ\copyright ½285\\ & =& \frac{1}{7}; 278â\copyright ½xâ\copyright ½285\end{array}$

When, no bolt-on trace elements are used.

The probability that the temperature will fall in range between

$\begin{array}{rcl}p\left(280<x<284\right)& =& \underset{280}{\overset{284}{∫}}\frac{1}{30}dx\\ & =& \frac{1}{30}{\left.x\right]}_{280}^{284}\\ & =& \frac{1}{30}\left(284-280\right)\\ & =& 0.133\\ & & \end{array}$

Hence, the probability is 0.133.

When, bolt-on trace elements are used

The probability that the temperature will fall in range between $280°Fand 284°F$

$\begin{array}{rcl}p\left(280<x<284\right)& =& \underset{280}{\overset{284}{∫}}\frac{1}{7}dx\\ & =& \frac{1}{7}{\left.x\right]}_{280}^{284}\\ & =& \frac{1}{7}\left(284-280\right)\\ & =& 0.571\end{array}$

Thus, the probability is0.571.

When, no bolt-on trace elements are used.

The probability of plastic plating is

$\begin{array}{rcl}p\left(xâ\copyright ½268\right)& =& \underset{260}{\overset{268}{∫}}\frac{1}{30}dx\\ & =& \frac{1}{30}{\left.x\right]}_{260}^{268}\\ & =& \frac{1}{30}\left(268-260\right)\\ & =& 0.267\end{array}$

Therefore, the probability is 0.267.

When, the bolt-on trace elements are used.

The probability of plastic plating is zeroas the value of x is not in between the range when the temperature is lower or$268°F$.