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Question Number 11095 by FilupS last updated on 12/Mar/17

You have a line of length 1. You place two random points on the line. What is the average distance between the two points? Please show your working.

$$\mathrm{You}\:\mathrm{have}\:\mathrm{a}\:\mathrm{line}\:\mathrm{of}\:\mathrm{length}\:\mathrm{1}. \\ $$$$\mathrm{You}\:\mathrm{place}\:\mathrm{two}\:\mathrm{random}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}. \\ $$$$\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{average}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{points}?\:\mathrm{Please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working}. \\ $$

Commented by FilupS last updated on 12/Mar/17

For context, the answer is (1/3)

$$\mathrm{For}\:\mathrm{context},\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Answered by ajfour last updated on 12/Mar/17

(1/4).

$$\frac{\mathrm{1}}{\mathrm{4}}.\: \\ $$

Commented by FilupS last updated on 12/Mar/17

incorrect please show working. it helps to see mistakes, etc.

$$\mathrm{incorrect} \\ $$$$\: \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{working}. \\ $$$$\: \\ $$$$\mathrm{it}\:\mathrm{helps}\:\mathrm{to}\:\mathrm{see}\:\mathrm{mistakes},\:\mathrm{etc}. \\ $$

Commented by ajfour last updated on 12/Mar/17

∫^(1 ) _0 {((∫_x ^1 (y−x)dy)/(1−x)) }dx

$$\underset{\mathrm{0}} {\int}^{\mathrm{1}\:} \left\{\frac{\int_{{x}} ^{\mathrm{1}} \left({y}−{x}\right){dy}}{\mathrm{1}−{x}}\:\right\}{dx} \\ $$

Commented by FilupS last updated on 12/Mar/17

Please explain where you derived this result?

$$\mathrm{Please}\:\mathrm{explain}\:\mathrm{where}\:\mathrm{you}\:\mathrm{derived} \\ $$$$\mathrm{this}\:\mathrm{result}? \\ $$

Answered by ajfour last updated on 12/Mar/17

∫_0 ^1 ∫_0 ^1 ∣y−x∣dy dx =(1/3) . help interpreting !

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \mid{y}−{x}\mid{dy}\:{dx}\:=\frac{\mathrm{1}}{\mathrm{3}}\:. \\ $$$${help}\:{interpreting}\:! \\ $$

Commented by FilupS last updated on 12/Mar/17

Although this gives the correct solution, I am unsure that this is a correct method of approaching the problem. The method I am aware of is via probability density

$$\mathrm{Although}\:\mathrm{this}\:\mathrm{gives}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{solution}, \\ $$$$\mathrm{I}\:\mathrm{am}\:\mathrm{unsure}\:\mathrm{that}\:\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{correct}\:\mathrm{method} \\ $$$$\mathrm{of}\:\mathrm{approaching}\:\mathrm{the}\:\mathrm{problem}. \\ $$$$\: \\ $$$$\mathrm{The}\:\mathrm{method}\:\mathrm{I}\:\mathrm{am}\:\mathrm{aware}\:\mathrm{of}\:\mathrm{is}\:\mathrm{via} \\ $$$$\mathrm{probability}\:\mathrm{density} \\ $$

Commented by FilupS last updated on 12/Mar/17

please tell me how you got to this result

$$\mathrm{please}\:\mathrm{tell}\:\mathrm{me}\:\mathrm{how}\:\mathrm{you}\:\mathrm{got}\:\mathrm{to} \\ $$$$\mathrm{this}\:\mathrm{result} \\ $$

Commented by ajfour last updated on 12/Mar/17

Commented by FilupS last updated on 12/Mar/17

Although you have gotten the same result, i think you have solved for a similar question where (x, y)→R^2 I beleive this question has (x, y)→R^1 I honestly don′t know The method I am looking for has to do with probability

$$\mathrm{Although}\:\mathrm{you}\:\mathrm{have}\:\mathrm{gotten}\:\mathrm{the}\:\mathrm{same}\:\mathrm{result}, \\ $$$$\mathrm{i}\:\mathrm{think}\:\mathrm{you}\:\mathrm{have}\:\mathrm{solved}\:\mathrm{for}\:\mathrm{a}\:\mathrm{similar} \\ $$$$\mathrm{question}\:\mathrm{where}\:\left({x},\:{y}\right)\rightarrow\mathbb{R}^{\mathrm{2}} \\ $$$$\: \\ $$$$\mathrm{I}\:\mathrm{beleive}\:\mathrm{this}\:\mathrm{question}\:\mathrm{has}\:\left({x},\:{y}\right)\rightarrow\mathbb{R}^{\mathrm{1}} \\ $$$$\: \\ $$$$\mathrm{I}\:\mathrm{honestly}\:\mathrm{don}'\mathrm{t}\:\mathrm{know} \\ $$$$\: \\ $$$$\mathrm{The}\:\mathrm{method}\:\mathrm{I}\:\mathrm{am}\:\mathrm{looking}\:\mathrm{for}\:\mathrm{has}\:\mathrm{to}\:\mathrm{do} \\ $$$$\mathrm{with}\:\mathrm{probability} \\ $$

Commented by ajfour last updated on 12/Mar/17

you can, as a separate matter view my proof to Q.11036.

$${you}\:{can},\:{as}\:{a}\:{separate}\:{matter} \\ $$$${view}\:{my}\:{proof}\:{to}\:{Q}.\mathrm{11036}. \\ $$

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