(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let's have a box in shape of a square(viewed from the top) from the corner of which a smaller square was cut out.The side of a bigger square is 2a, side of the smaller square is a long.

We've got evenly distributed corn seeds all over the box,randomly selected seed is defined by coordinates [itex]x,y \in [0,2a][/itex]

2. Relevant equations

a.) Write down the combined probability distribution for [itex]w(x,y)[/itex]

b.) Write down the projected probability distribution for [itex]u(x)[/itex](independent of [itex]y[/itex])

c.) calculate the correlation coefficient [itex]r_{x,y}[/itex]

3. The attempt at a solution

a.) [itex]1/3a^₂[/itex]

b.) [itex]u(x)= 1/3a[/itex] if [itex]x \in [0,a][/itex]

[itex]u(x) = 2/3a[/itex] if [itex]x \in [a,2a][/itex]

c.) since [itex]r_{x,y} =\frac{\sigma_{x,y}}{\sigma_{x} \sigma_{y}} [/itex], i calculated each variance seperately:

[itex]\sigma_{x} = \int xu(x)dx[/itex]

[itex]\sigma_{y} = \int yu(y)dx[/itex]

[itex]\sigma_{x,y} = \int\int (x - \overline{x})(y - \overline{y})w(x,y)dxdy[/itex]

Is that right?

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# Combined probability distribution

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