Suppose three random variables X, Y, Z have a joint distribution PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). Then X and Y...
Suppose three random variables X, Y, Z have a joint distribution
| PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). |
Then X and Y are independent given Z? True or False
Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form
| PX,Y,Z(x,y,z)=h(x,z)g(y,z), |
where h,g are some functions? True or false
Homework Answers
1.
False
As the joint distribution includes term PZ|X(z|x) that means distribution of Z is dependent on X and also PY|Z(y|z) that means Y is dependent on Z, so by transitivity, Y is dependent on X hence they are not independent given Z.
2.
True.
The X and Y are dependent on Z only. And the joint pdf can be written in form of product x|z and y|z.
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Suppose three random variables X, Y, Z have a joint
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