Which function represents the marginal probability density of X?

Which function represents the marginal probability density of X?

• A. Pr(X=x)
• B. ∫ fxy(x,y) dy
• C. f(x) = fx(x,y)/fy(y)
• D. ∑ fxy(x,y)

Answer: (B)

The marginal probability density function of a random variable \(X\) can be obtained by integrating the joint probability density function \(f_{XY}(x,y)\) over all possible values of the other variable \(Y\). This operation is carried out to isolate the influence of \(X\) and provides the probability distribution of \(X\) alone, regardless of \(Y\). Integrating \(f_{XY}(x,y)\) with respect to \(y\), denoted as \(\int f_{XY}(x,y) \, dy\), effectively sums up the contributions from all possible outcomes of \(Y\) for each specific value of \(X\). This results in the marginal probability density function \(f_X(x)\) for \(X\), which reflects the likelihood of \(X\) occurring independently of \(Y\). The other choices do not properly reflect the formulation needed to derive the marginal probability density function of \(X\). The first option, \(Pr(X=x)\), represents the probability at a specific point which is not applicable for continuous random variables. The third choice gives a formula that involves conditioning on \(Y\), which is not what is needed for marginalization. The fourth choice involves summation, which

The marginal probability density function of a random variable (X) can be obtained by integrating the joint probability density function (f_{XY}(x,y)) over all possible values of the other variable (Y). This operation is carried out to isolate the influence of (X) and provides the probability distribution of (X) alone, regardless of (Y).

Integrating (f_{XY}(x,y)) with respect to (y), denoted as (\int f_{XY}(x,y) , dy), effectively sums up the contributions from all possible outcomes of (Y) for each specific value of (X). This results in the marginal probability density function (f_X(x)) for (X), which reflects the likelihood of (X) occurring independently of (Y).

The other choices do not properly reflect the formulation needed to derive the marginal probability density function of (X). The first option, (Pr(X=x)), represents the probability at a specific point which is not applicable for continuous random variables. The third choice gives a formula that involves conditioning on (Y), which is not what is needed for marginalization. The fourth choice involves summation, which