Statistics Help-Joint pmf and marginal pmfs

In summary, the joint pmf of Y=max(X1,X2) and W=min(X1, X2) can be found by considering the 2D lattice formed by the two iid discrete uniform random variables X1 and X2. Each lattice point has equal probability, and by considering a few cases, we can see that the formula for the joint pmf is (k-1)/n^2, where k is the value of Y and n is the support of the random variables. The marginal pmf's of W and Y can be found by summing the joint pmf over the appropriate values of X1 and X2. This approach can be visualized by plotting the
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runner4life
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Statistics Help--Joint pmf and marginal pmfs

Homework Statement



Let X1 and X2 be iid discrete uniform random variables with support on the integers {1,2,3,...,n}.

Find the joint pmf of Y=max(X1,X2) and W=min(X1, X2).
Find the marginal pmf's of W and Y

Homework Equations





The Attempt at a Solution


I need help on how to set this problem up. the max and min is throwing me off.
 
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  • #2


so imagine the variables as forming a 2D lattice, each of the lattice points having equal probability

as always consdier a few cases to get going
max(x1,x2) = 1, can only be (1,1)
max(x1,x2) = 2, can be (2,1), (2,2), (1,2)
max(x1,x2) = 3, can be (3,1), (3,2) (3,3), (3,1), (3,2)

this form should give you a hint on how to work out a generic formula

to understand why this formula works, try plotting the points on your lattic & see where they fall
 

1. What is the difference between joint pmf and marginal pmfs?

Joint pmf (probability mass function) refers to the probability of two or more random variables occurring simultaneously. On the other hand, marginal pmf refers to the probability distribution of a single random variable without considering the other variables. In simple terms, joint pmf deals with multiple variables, while marginal pmf deals with a single variable.

2. How do you calculate the joint pmf of two random variables?

To calculate the joint pmf, you need to multiply the probabilities of each individual variable occurring at the same time. For example, if the probabilities of x and y are 0.5 and 0.3 respectively, the joint pmf would be 0.5 x 0.3 = 0.15.

3. Can you give an example of using joint pmf and marginal pmfs in real life?

One example is in medical studies, where scientists may use joint pmf to determine the probability of two treatments having a successful outcome for a patient. Marginal pmfs can also be used to analyze the effectiveness of each treatment separately.

4. How can joint pmf and marginal pmfs help in decision making?

By understanding the probability of multiple variables occurring together, joint pmf can help in making more informed decisions. Marginal pmfs can also provide valuable insights into the individual probabilities of each variable, aiding in decision making.

5. Is it possible for two variables to have a joint pmf but not have a marginal pmf?

Yes, it is possible. If the two variables are not independent, meaning that the probability of one variable changes based on the value of the other, then a joint pmf can be calculated. However, in this case, there may not be a well-defined marginal pmf for each variable separately.

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