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# Marginal Probability Distribution, independent variables, expected value

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Time spent by two clerks is defined by

f(x,y) = { x+y if 0&#8804;x &#8804;1 ; 0&#8804;y&#8804;1
{ 0 else

a) Find marginal probability distribution f1(x) and f2(y)
b) Are two random variable x,y independent? why?
c) Are x,y correlated? why?
d) Suppose proportion "d" of dead time (time when no assigned duties are being performed) for two clerks is given by the relation d=1-[(x+y)/2]. find E(d), the expected value of d.

https://brainmass.com/statistics/probability/marginal-probability-distribution-independent-variables-expect-296924

#### Solution Preview

Time spent by two clerks is defined by

f(x,y) = { x+y if 0≤x ≤1 ; 0≤y≤1
{ 0 else

a)find marginal probability distribution f1(x) and f2(y)
f1(x)=∫_0^1▒〖f(x,y)dy=∫_0^1▒〖(x+y)dy=x∫_0^1▒〖dy+∫_0^1▒〖(y)dy=x+〗(1/2)〗〗〗 if 0≤x ≤1, 0 else
f2(y)=∫_0^1▒〖f(x,y)dx=∫_0^1▒〖(x+y)dx=∫_0^1▒〖xdx+∫_0^1▒〖(y)dx=y+〗(1/2)〗〗〗 if 0≤y ≤1, ...

#### Solution Summary

The marginal probability distribution, and independent variables are determined.

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