Please help !! Prob and stats

The lifetime of a certain type of washing machine is exponentially distributed, with a mean of m years. the manu facturer offers a guarantee on every machine sold: if the machine breaks down irreparably within the first year of use( so that its lifetime is less than one year), it will be replaced free of charge.

(a) what proportion of machines need to be replaced free of charge?

(b)any replacement machine that fails within the first year of its own lifetime will also be replaced free of charge. the lifetimes of successive replacements are mutually independent. for every machine sold, how many replacements does the company expect to have to provide?

(c)the cost manufaturing and supplying a washing machine is £200, and the machines sell for £500 each. If the mean lifetime is 5 years, how much profit dows the manufacturer expect to make per machine sold?

(d)It is suggested that the manufacturer may be able to increase profitability by making the machine more reliable, so that fewer replacements are required. Howeverm this will increase produciton costs. Specificallym the cost (in £) of manufacturing and supplying a machine with a mean lifetime of m years is C(m)=180+4m. The manufacturer wishes to keep the selling price fixed at £500 per machine. what value of m is maximises the expected profit?

(e)Briefly, explaine why the exponential distribution might, as a first approximation. provide a resonable model for the distribution of washing machine lifetime iin practice. Does this model have any unrealistic features?

Quote:

---

Originally Posted by kevek

The lifetime of a certain type of washing machine is exponentially distributed, with a mean of m years. the manu facturer offers a guarantee on every machine sold: if the machine breaks down irreparably within the first year of use( so that its lifetime is less than one year), it will be replaced free of charge.

(a) what proportion of machines need to be replaced free of charge?

(b)any replacement machine that fails within the first year of its own lifetime will also be replaced free of charge. the lifetimes of successive replacements are mutually independent. for every machine sold, how many replacements does the company expect to have to provide?

[snip]

---

To get you started, here are hints for the first couple:

(a) Let X be the random variable lifetime of machine.

Calculate $\displaystyle \Pr(X < 1) = \int_{0}^{1} f(x) \, dx = p$.

(b) Let Y be the random variable number of machines that break down within one year.

$\displaystyle E(Y) = \sum_{i=0}^{\infty} i p^i$.

Quote:

---

Originally Posted by kevek

The lifetime of a certain type of washing machine is exponentially distributed, with a mean of m years. the manu facturer offers a guarantee on every machine sold: if the machine breaks down irreparably within the first year of use( so that its lifetime is less than one year), it will be replaced free of charge.

[snip]

(c)the cost manufaturing and supplying a washing machine is £200, and the machines sell for £500 each. If the mean lifetime is 5 years, how much profit dows the manufacturer expect to make per machine sold?

(d)It is suggested that the manufacturer may be able to increase profitability by making the machine more reliable, so that fewer replacements are required. Howeverm this will increase produciton costs. Specificallym the cost (in £) of manufacturing and supplying a machine with a mean lifetime of m years is C(m)=180+4m. The manufacturer wishes to keep the selling price fixed at £500 per machine. what value of m is maximises the expected profit?

(e)Briefly, explaine why the exponential distribution might, as a first approximation. provide a resonable model for the distribution of washing machine lifetime iin practice. Does this model have any unrealistic features?

---

(c) $\displaystyle E(P) = (300)(1 - p) + (100)(p) + (-100) p^2 + (-300) p^3 + (-500) p^4 + .... $

(d) Generalise the logic behind part (c)

(e) Research for you to do .....

Thank you very much.

Quote:

---

Originally Posted by mr fantastic

(c) $\displaystyle E(P) = (300)(1 - p) + (100)(p) + (-100) p^2 + (-300) p^3 + (-500) p^4 + .... $

(d) Generalise the logic behind part (c)

(e) Research for you to do .....

---

why does part (c) has infinite polynomial?

Quote:

---

Originally Posted by kevek

why does part (c) has infinite polynomial?

---

Because there's a finite probability that the number of machines that break down in one year is 1, 2, 3, 4, ......

You can get this sum by playing tricks with the infinite geometric series $\displaystyle \sum_{i=1}^{\infty} p^i = \frac{p}{1 - p}$ ....