a) The base chassis of a manufactured product is made up of numerous components. There can sometimes be differences in dimensions between components, requiring the partially assembled product to be sent to another department for extra fitting work because some components cannot be fitted in place. The majority of fitting problems are caused by two particular components (identified as B and F): the chances of a randomly chosen component of type B not fitting are 3.7%, and the chances of a randomly chosen component of type F not fitting are 2.8%. (i) Given that the problems with fitting these two components are independent of each other and that components are selected randomly for assembly on each base chassis, what is the probability that both components B and F will have problems fitting on a given chassis? ii) How likely is it that there will be problems fitting components B and F to a given chassis under the same assumptions? (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. 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. c) An electrical component's failure ages follow an exponential distribution with an average failure rate of λ = 0.85 failures per year. Determine what is the probability that a randomly selected component will fail between three and five years after it is installed.
a) The base chassis of a manufactured product is made up of numerous components. There can sometimes be differences in dimensions between components, requiring the partially assembled product to be sent to another department for extra fitting work because some components cannot be fitted in place. The majority of fitting problems are caused by two particular components (identified as B and F): the chances of a randomly chosen component of type B not fitting are 3.7%, and the chances of a randomly chosen component of type F not fitting are 2.8%.
(i) Given that the problems with fitting these two components are independent of each other and that components are selected randomly for assembly on each base chassis, what is the
ii) How likely is it that there will be problems fitting components B and F to a given chassis under the same assumptions?
(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.
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.
c) An electrical component's failure ages follow an exponential distribution with an average failure rate of λ = 0.85 failures per year. Determine what is the probability that a randomly selected component will fail between three and five years after it is installed.
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