What is the conditional probability function of X given Y=y for a multivariate distribution?

What is the conditional probability function of X given Y=y for a multivariate distribution?

• A. fx(x) + fy(y)
• B. fxy(x,y)/fy(y)
• C. fxy(x|Y=y)
• D. fx(x|Y=y) + fy(y)

Answer: (B)

The conditional probability function of a random variable \( X \) given another random variable \( Y \) at a specific value \( y \) is defined using the joint probability density function and the marginal probability density function of \( Y \). The correct formulation is given by the ratio of the joint density function \( f_{XY}(x,y) \) to the marginal density function \( f_Y(y) \). Mathematically, this is expressed as: \[ f_{X|Y}(x|Y=y) = \frac{f_{XY}(x,y)}{f_Y(y)} \] This formula captures how likely \( X \) is to take a value \( x \) when we know that \( Y \) equals \( y \). The joint density function \( f_{XY}(x,y) \) describes the probability of \( X \) and \( Y \) occurring together, while \( f_Y(y) \) ensures that we are scaling this probability correctly by the likelihood of \( Y \) taking on the value \( y \). The other choices do not accurately represent the definition of conditional probability. For example, adding densities or functions does not yield a meaningful probability distribution, as seen in options that combine or

The conditional probability function of a random variable ( X ) given another random variable ( Y ) at a specific value ( y ) is defined using the joint probability density function and the marginal probability density function of ( Y ). The correct formulation is given by the ratio of the joint density function ( f_{XY}(x,y) ) to the marginal density function ( f_Y(y) ).

Mathematically, this is expressed as:

[

f_{X|Y}(x|Y=y) = \frac{f_{XY}(x,y)}{f_Y(y)}

]

This formula captures how likely ( X ) is to take a value ( x ) when we know that ( Y ) equals ( y ). The joint density function ( f_{XY}(x,y) ) describes the probability of ( X ) and ( Y ) occurring together, while ( f_Y(y) ) ensures that we are scaling this probability correctly by the likelihood of ( Y ) taking on the value ( y ).

The other choices do not accurately represent the definition of conditional probability. For example, adding densities or functions does not yield a meaningful probability distribution, as seen in options that combine or