**Introduction**

In easy words, we can say about marginal and conditional distributions, the marginal probability is the likelihood of a single event occurring in the absence of any other events. A conditional probability, on the other hand, is the likelihood that an event will occur if another event has already occurred.

## Conditional distribution

A conditional distribution is a random variable’s probability distribution determined using conditional probability rules after observing the realization of another random variable.

For example, the following table shows the results of a survey that asked 100 people which sport they like the most: basketball, badminton, or football.

Badminton | Basketball | Football | Total | |

Male | 12 | 16 | 20 | 48 |

Female | 22 | 14 | 16 | 52 |

Total | 34 | 30 | 36 | 100 |

If we want to know the probability that a man prefers a particular sport, then this is an example of conditional probability.

The value of one random variable is known (the person is male), but the value of the other random variable is unknown (we don’t know their favorite sport).

We may find the conditional distribution of male sports preferences by looking at the values in the Male row in the table.

Badminton | Basketball | Football | Total | |

Female | 22 | 14 | 16 | 52 |

The conditional distribution be calculated as:

- Females who likes Badminton: 22/52 = 0.423
- Females who likes Basketball: 14/52 = 0.269
- Females who likes Football: 16/52 = 0.307

The sum of the probabilities adds up to: 22/52 + 14/52 + 16/52 = 48/48 = 1

We can use conditional distributions to answer questions like : What are the chances that a person’s favorite sport is badminton if they are basketball?

From the conditional distribution we calculated earlier, we can see the probability is 0.423 .

When we calculate a conditional distribution, we’re expressing our interest in a certain subset of the entire population. Females were the subpopulation in the prior example.

And we indicate we’re interested in a particular character of interest when we wish to calculate a probability for this subpopulation. In the preceding example, the character of interest was badminton.

We simply divide the value of the character of interest by the total values in the subpopulation to find the probability that the character of interest appears in the subpopulation.

Marginal distribution

In a contingency table, a marginal distribution is a frequency or relative frequency distribution of either the row or column variable. In a contingency table, a conditional distribution lists the relative frequency of each category of the response variable for a given value of the explanatory variable.

The distribution of each of these separate variables is called a marginal distribution. The marginal distributions in this table are shown in the table’s margins.

For example, the following table shows the results of a survey that asked 100 people which subjects they like the most: mathematics, physics or chemistry.

Mathematics | Physics | Chemistry | Total | |

Male | 11 | 17 | 20 | 48 |

Female | 23 | 15 | 14 | 52 |

Total | 34 | 32 | 34 | 100 |

For example we would say that:

- Mathematics: 34
- Physics: 32
- Chemistry: 34

We could also write the marginal distribution of sports in percentage terms:

- Mathematics: 34 %
- Physics: 32 %
- Chemistry: 34 %

And we would say that the marginal distribution of gender is:

- Male: 48 %
- Female: 52 %

One can do marginal probability examples with solutions. They are easy and same like the upper example but in this we have to find probability.

What is the difference between Marginal and conditional distribution?

If X and Y are two random variables, the marginal distribution of X is the univariate pdf of X, and the marginal distribution of Y is the univariate pdf of Y. As a result, when you encounter the word marginal, think of the distribution of a single data series. Don’t be misled into thinking that marginal signifies something other than a univariate (single variable) assessment.

In the case of conditional distribution, we do a bivariate (two variable) assessment, but we evaluate the relationship of the univariate components to one another: The distribution of X conditional on or given the recognition of Y’s data is known as conditional pdf. The concept is that an observation in X corresponds to a similarly positioned observation in Y, and therefore we think of X in relation to what is observed in Y. To put it another way, the conditional pdf is a weak way of defining the distribution of X as a function of Y.

**Conclusion:**

From the above data, we conclude that marginal distributions and conditional distributions both are useful in the statistical world like surveys, etc.