MHB Components working at a particular time

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The discussion centers on calculating the expected number of working components in a system of 10, where each component j has a working probability of 1/j. The expected value is derived using the formula E{n} = H_{10}, which represents the 10th harmonic number. Participants note that the independence of the components means their probabilities do not influence each other. The final expected value indicates how many components are likely functioning at any given time. This approach effectively combines probability theory with practical application in systems analysis.
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Hello .Sorry for all the questions I am asking...but I have another one I would like to seek help for.:

Suppose a system has 10 components and that a particular time the jth component is working with probability 1/j for j= 1,2,...10. How many components do you expect to be working at that particular time?

All I have done so far is calculating the probability for all 10 components and I know that the sum of all possible probabilities is 1.

But how should I solve the question?
 
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oyth94 said:
Hello .Sorry for all the questions I am asking...but I have another one I would like to seek help for.:

Suppose a system has 10 components and that a particular time the jth component is working with probability 1/j for j= 1,2,...10. How many components do you expect to be working at that particular time?

All I have done so far is calculating the probability for all 10 components and I know that the sum of all possible probabilities is 1.

But how should I solve the question?

The expected number of working components is...

$\displaystyle E\{n\} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{10} = H_{10}\ (1)$ Since the status of components are independent r.v. nothing can be said about the sum of their probabilities... Kind regards $\chi$ $\sigma$
 
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