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
(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
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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