Marginal/Conditional Probability Mass Functions

In summary, a bag contains four dice labelled 1,...,4, with each die having a different number of white and black faces. One die is chosen at random and rolled, and X is defined as the number on the chosen die. Y is defined as 0 if the face showing on the die is black and 1 if it is white. To display the values of the marginal pmf for X, we use the equation Px(x) = P(X=x) = \sum P(x,y). For the conditional pmf of Y given X=x, we use the equation PY/X(y,x) = P(x,y)/Px(x). The resulting tables will have values for each possible value of X, with corresponding probabilities for Y
  • #1
vikkisut88
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Homework Statement


A bag contains four dice labelled 1,...,4. The die labelled j has j white faces and (6-j) black faces, j = 1,...,4. A die is chosen at random from the bag and rolled. We define X = the number labelling the chosen die.
Y = {0 if the face showing on the die is black; 1 if the face showing on the die is white.

Construct a table displaying the values of the marginal pmf (probability mass function) for X and a separate table displaying the values of the conditional pmf for Y fiven X=x for general x.

Homework Equations


Marginal pmf for X is Px(x) = P(X=x) = [tex]\sum[/tex] P(x,y)
Conditional pmf for Y given X=x is PY/X(y,x) = P(x,y)/Px(x)

The Attempt at a Solution


Okay so i know the two equations above for the two pmf's. I'm presuming that for the first part (marginal pmf) that you just use x=1,2,3,4 and that Px(x) is 1/4 for each value of x?
For the second table I'm not so sure as I don't know which y values to put into my table, and how to work out any probabilities as it has to be for a general x and not a given value of x. As at the beginning of the question it states that y=0 or y=1 is it just these two values that i put into my table n then depending on when x=1,...,4 will alter the probability
 
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  • #2
vikkisut88 said:
as it has to be for a general x and not a given value of x
I think they're asking for a 2D table, so for each x you have a distribution for y.
 
  • #3
X=1, then Pr(Y=0|X = 1)=5/24, Pr(Y=1|X=1)=1/24;
X=2, then Pr(Y=0|X=2)=4/24, Pr(Y=1|X=2)=2/24;
Continue for each value of X.
 

FAQ: Marginal/Conditional Probability Mass Functions

What is a marginal probability mass function?

A marginal probability mass function is a type of probability distribution that shows the probability of a single random variable taking on a specific value. It is obtained by summing or integrating over all other random variables in a multivariate distribution. In simpler terms, it shows the probability of a single event occurring without taking into consideration any other events.

How is a marginal probability mass function different from a conditional probability mass function?

A marginal probability mass function only considers the probability of a single event occurring, while a conditional probability mass function takes into account the probability of that event occurring given that another event has already occurred. In other words, a marginal probability mass function shows the overall probability of an event occurring, while a conditional probability mass function shows the probability of an event occurring under specific conditions.

How can marginal and conditional probability mass functions be used in real-world applications?

Marginal and conditional probability mass functions are commonly used in data analysis, machine learning, and other fields where probability and statistics play a role. They can help in making predictions, identifying patterns and trends, and understanding the relationship between different variables in a dataset.

What is the difference between a probability mass function and a probability density function?

The main difference is that a probability mass function is used for discrete random variables, while a probability density function is used for continuous random variables. A probability mass function assigns probabilities to specific values of a discrete random variable, while a probability density function assigns probabilities to intervals of values for a continuous random variable.

How can you calculate the marginal probability mass function from a joint probability mass function?

To calculate the marginal probability mass function from a joint probability mass function, you need to sum or integrate over all possible values of the other random variables. This means that you add up the probabilities of all the outcomes for the random variable of interest, while keeping the other random variables fixed. The resulting values will give you the probabilities for each possible value of the random variable of interest.

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