Marginal distribution

Given a known joint distribution of two discrete random variables, say, X and Y, the marginal distribution of either variable – X for example – is the probability distribution of X when the values of Y are not taken into consideration. This can be calculated by summing the joint probability distribution over all values of Y. Naturally, the converse is also true: the marginal distribution can be obtained for Y by summing over the separate values of X.

${\displaystyle p_{X}(x_{i})=\sum _{j}p(x_{i},y_{j})}$, and ${\displaystyle p_{Y}(y_{j})=\sum _{i}p(x_{i},y_{j})}$

A marginal probability can always be written as an expected value:

$${\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\operatorname {E} _{Y}[p_{X\mid Y}(x\mid y)]\;.}$$

Intuitively, the marginal probability of X is computed by examining the conditional probability of X given a particular value of Y, and then averaging this conditional probability over the distribution of all values of Y.

This follows from the definition of expected value (after applying the law of the unconscious statistician)

$${\displaystyle \operatorname {E} _{Y}[f(Y)]=\int _{y}f(y)p_{Y}(y)\,\mathrm {d} y.}$$

Therefore, marginalization provides the rule for the transformation of the probability distribution of a random variable Y and another random variable X=g(Y):

$${\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\int _{y}\delta {\big (}x-g(y){\big )}\,p_{Y}(y)\,\mathrm {d} y.}$$

Given two continuous random variables X and Y whose joint distribution is known, then the marginal probability density function can be obtained by integrating the joint probability distribution, f, over Y, and vice versa. That is

${\displaystyle f_{X}(x)=\int _{c}^{d}f(x,y)\,dy}$

${\displaystyle f_{Y}(y)=\int _{a}^{b}f(x,y)\,dx}$

where ${\displaystyle x\in [a,b]}$, and ${\displaystyle y\in [c,d]}$.

Finding the marginal cumulative distribution function from the joint cumulative distribution function is easy. Recall that:

• For discrete random variables,
  $${\displaystyle F(x,y)=P(X\leq x,Y\leq y)}$$

  
• For continuous random variables,
  $${\displaystyle F(x,y)=\int _{a}^{x}\int _{c}^{y}f(x',y')\,dy'dx'}$$

  

If X and Y jointly take values on [a, b] × [c, d] then

${\displaystyle F_{X}(x)=F(x,d)}$ and ${\displaystyle F_{Y}(y)=F(b,y)}$

If d is ∞, then this becomes a limit ${\textstyle F_{X}(x)=\lim _{y\to \infty }F(x,y)}$. Likewise for ${\displaystyle F_{Y}(y)}$.

The marginal probability is the probability of a single event occurring, independent of other events. A conditional probability, on the other hand, is the probability that an event occurs given that another specific event has already occurred. This means that the calculation for one variable is dependent on another variable.[2]

The conditional distribution of a variable given another variable is the joint distribution of both variables divided by the marginal distribution of the other variable.[3] That is,

• For discrete random variables,
  $${\displaystyle p_{Y|X}(y|x)=P(Y=y\mid X=x)={\frac {P(X=x,Y=y)}{P_{X}(x)}}}$$

  
• For continuous random variables,
  $${\displaystyle f_{Y|X}(y|x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}}$$

  

Suppose there is data from a classroom of 200 students on the amount of time studied (X) and the percentage of correct answers (Y).[4] Assuming that X and Y are discrete random variables, the joint distribution of X and Y can be described by listing all the possible values of p(x_i,y_j), as shown in Table.3.

The marginal distribution can be used to determine how many students scored 20 or below: ${\displaystyle p_{Y}(y_{1})=P_{Y}(Y=y_{1})=\sum _{i=1}^{4}P(x_{i},y_{1})={\frac {2}{200}}+{\frac {8}{200}}={\frac {10}{200}}}$, meaning 10 students or 5%.

The conditional distribution can be used to determine the probability that a student that studied 60 minutes or more obtains a scored of 20 or below: ${\displaystyle p_{Y|X}(y_{1}|x_{4})=P(Y=y_{1}|X=x_{4})={\frac {P(X=x_{4},Y=y_{1})}{P(X=x_{4})}}={\frac {8/200}{70/200}}={\frac {8}{70}}={\frac {4}{35}}}$, meaning there is about a 11% probability of scoring 20 after having studied for at least 60 minutes.