1. Let's start by stating the problem: constructing a conditional distribution of a random variable $Y$ given another variable $X=x$. 2. The conditional distribution of $Y$ given $X=x$ is defined as the probability distribution of $Y$ when $X$ is fixed at $x$. 3. The formula for the conditional probability mass function (pmf) or probability density function (pdf) is: $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}$$ where $f_{X,Y}(x,y)$ is the joint distribution of $X$ and $Y$, and $f_X(x)$ is the marginal distribution of $X$. 4. Important rules: - The marginal distribution $f_X(x)$ must be positive (i.e., $f_X(x) > 0$) for the conditional distribution to be defined. - The conditional distribution $f_{Y|X}(y|x)$ is a valid probability distribution over $y$ for each fixed $x$. 5. To construct the conditional distribution: - Find the joint distribution $f_{X,Y}(x,y)$. - Find the marginal distribution $f_X(x)$ by summing or integrating over $y$: $$f_X(x) = \sum_y f_{X,Y}(x,y) \quad \text{(discrete)}$$ or $$f_X(x) = \int f_{X,Y}(x,y) \, dy \quad \text{(continuous)}$$ - Compute the conditional distribution using the formula above. 6. Example: Suppose $X$ and $Y$ are discrete with joint pmf $f_{X,Y}(x,y)$. To find $f_{Y|X}(y|x)$: - Calculate $f_X(x) = \sum_y f_{X,Y}(x,y)$. - Then compute $f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}$. This gives the distribution of $Y$ when $X$ is fixed at $x$. 7. In summary, constructing a conditional distribution involves dividing the joint distribution by the marginal distribution of the conditioning variable, ensuring the marginal is positive, and interpreting the result as a probability distribution over the conditioned variable.