Conditional density formula for probability density function
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I know thatfX|Y(x|y)=fX,Y(x,y)fY(y)Is defined to be the conditional density of function of X given Y.
However, I have yet to see how the conditional probability function of several variables is defined. If I had to guess, I would say it is defined like this:fX,Y|Z(x,y|z)=fX,Y,Z(x,y,z)fZ(z)And hopefully it is also true that:
fX|Y,Z(x|y,z)=fX,Y,Z(x,y,z)fY,Z(y,z)For more variables, the pattern is clear, and I would understand if this was the definition.
My biggest problem is the following:
If X and Y are independent uniformly distributed on (−1,1) random variables, what is the conditional density function of X and Y given that X2+Y2≤1?
Intuitively and logically, I know that the answer is 1π, but I would like to find this answer through the use of the above formulas and definitions I thought about P(X≤a,Y≤b|X2+Y2≤1), intending to differentiate after I had gotten the integral of the density function... of what density function? It cannot be 14 because where
would the condition fall? I could maybe define a new density function of X,Y,R=X2+Y2, but how would I obtain it?
My main question is: how do I get the conditional density using a more formal argument (i.e. using definitions and formulas)?
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However, I have yet to see how the conditional probability function of several variables is defined. If I had to guess, I would say it is defined like this:fX,Y|Z(x,y|z)=fX,Y,Z(x,y,z)fZ(z)And hopefully it is also true that:
fX|Y,Z(x|y,z)=fX,Y,Z(x,y,z)fY,Z(y,z)For more variables, the pattern is clear, and I would understand if this was the definition.
My biggest problem is the following:
If X and Y are independent uniformly distributed on (−1,1) random variables, what is the conditional density function of X and Y given that X2+Y2≤1?
Intuitively and logically, I know that the answer is 1π, but I would like to find this answer through the use of the above formulas and definitions I thought about P(X≤a,Y≤b|X2+Y2≤1), intending to differentiate after I had gotten the integral of the density function... of what density function? It cannot be 14 because where
would the condition fall? I could maybe define a new density function of X,Y,R=X2+Y2, but how would I obtain it?
My main question is: how do I get the conditional density using a more formal argument (i.e. using definitions and formulas)?
PLS MARK ME AS BRAINLIEST
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