How Do You Calculate the Conditional Distribution P(X = x | Y = y)?

In summary, the problem involves randomly selecting two cards from two different decks. The first deck has N cards labeled 1,2,...,N and the second deck has X cards labeled 1,2,...,X, where X depends on the value of the first card chosen. The question is asking for the conditional distribution of X given Y = y, where Y is the value of the second card chosen. The equation used to solve this problem is P(X = x | Y = y) = \frac{P(X = x , Y = y)}{P(Y = y)}. Further clarification is needed on how to find P(X = x, Y = y). Any assistance would be greatly appreciated.
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Homework Statement



A card is picked at random from N cards labeled 1,2,...,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,..., X and its number is Y. Find the conditional distribution of X given Y = y.

Homework Equations



[tex]P(X = x | Y = y) = \frac{P(X = x , Y = y)}{P(Y = y)}[/tex]


The Attempt at a Solution



From what I understand, there are two decks. The first deck has N cards, while the second deck has X cards, which depends on the value of the card chosen from the first deck, hence x and y are not independent.

I know that [tex]P(Y = y) = \frac{1}{x}[/tex]

I'm not sure how to find P(X = x , Y = y).

Any help would be greately appreciated. Thank you.
 
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  • #2
What is the value of P(X=x)? What does P(x,y) mean in general?
 

Related to How Do You Calculate the Conditional Distribution P(X = x | Y = y)?

What is a conditional distribution?

A conditional distribution is a probability distribution that shows the likelihood of an event occurring given that another event has already occurred. It is used to analyze the relationship between two variables and can provide insights into how one variable affects the other.

How is a conditional distribution different from a joint distribution?

A joint distribution shows the probability of two events occurring simultaneously, while a conditional distribution only considers the probability of one event occurring given the occurrence of another event. In other words, a joint distribution shows the relationship between two variables, while a conditional distribution shows the relationship between them in a specific scenario or condition.

How is a conditional distribution calculated?

A conditional distribution is calculated by dividing the joint probability of two events by the probability of the event that is already known to have occurred. This can be written as P(A|B) = P(A and B)/P(B), where A and B are two events.

What is the purpose of a conditional distribution in statistical analysis?

The purpose of a conditional distribution is to help understand the relationship between two variables and to make predictions based on that relationship. It can also be used to identify patterns or trends and to determine the probability of an event occurring given certain conditions.

What is an example of a real-world application of conditional distribution?

An example of a real-world application of conditional distribution is in market research. Companies may use it to analyze the relationship between customer demographics (such as age or income) and their purchasing behavior. By understanding the conditional distribution, companies can tailor their marketing strategies to specific groups of customers and increase their chances of success.

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