The discrete random variable \(X\) has probability distribution given by
$$\mathrm { P } ( X = x ) = \begin{cases} \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } & x = 0,1,2 , \ldots 0 & \text { otherwise } \end{cases}$$
Show that \(\mathrm { E } ( X ) = \lambda\) and that \(\operatorname { Var } ( X ) = \lambda\).
In light-weight chain, faults occur randomly and independently, and at a constant average rate of 0.075 per metre.
Calculate the probability that there are no faults in a 10 -metre length of this chain.
Use a distributional approximation to estimate the probability that, in a 500 -metre reel of light-weight chain, there are:
(A) fewer than 30 faults;
(B) at least 35 faults but at most 45 faults.
As part of an investigation into the quality of a new design of medium-weight chain, a sample of fifty 10 -metre lengths was selected.
Subsequent analysis revealed a total of 49 faults.
Assuming that faults occur randomly and independently, and at a constant average rate, construct an approximate \(98 \%\) confidence interval for the average number of faults per metre. [0pt]
[6 marks]
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