b) (i) The probability distribution of the number of customers visiting a specialty store each hour, N, is as follows: n    Pr ( N = n ) 5    0.1785 6    k 7    0.3105 8    0.1115 9    0.015 Find the value of k which makes this a valid distribution. (ii) In a certain mechanical component, the time to failure T (in hours) is distributed with the probability density function f(t) = 2.5 t^−3.5 for t ≥ 1. Calculate the mean E(T) of this variable. (iii) In the case of part (ii), E(T^2 ) = 5. Calculate Var(T) of the variable T based on this information

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(b)

(i) The probability distribution of the number of customers visiting a specialty store each hour, N, is as follows:

n    Pr ( N = n )

5    0.1785

6    k

7    0.3105

8    0.1115

9    0.015

Find the value of k which makes this a valid distribution.

(ii) In a certain mechanical component, the time to failure T (in hours) is distributed with the probability density function f(t) = 2.5 t^−3.5 for t ≥ 1.

Calculate the mean E(T) of this variable.

(iii) In the case of part (ii), E(T^2 ) = 5. Calculate Var(T) of the variable T based on this information.

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(c)The failure ages of a particular electrical component are observed to follow an exponential distribution with an average failure rate of λ = 0.85 failures per year. Find the probability that a randomly chosen component fails between three and five years after being placed in service

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