Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Let's start with the first example involving a discrete joint probability mass function. We have probabilities for outcomes defined for pairs (X,Y): P(X=x,Y=y) = 1/8 for the pairs (0,0), (0,1), (1,0), (1,1). What do you think we need to do first?
We should calculate the marginal distributions.
Exactly! To find the marginal distribution of X, we sum the probabilities of all occurrences where X takes a specific value. Can anyone tell me what P(X=0) would be?
It's P(0,0) plus P(0,1), which gives us 1/8 + 1/8 = 1/4.
Right! Now let's calculate P(Y=0). Who can assist with that?
That would be P(0,0) plus P(1,0), which is also 1/8 + 1/8 = 1/4.
Fantastic! Now, how can we check if X and Y are independent?
We need to see if P(0,0) equals P(X=0) times P(Y=0), right?
Absolutely! In this case, since P(0,0) is 1/8 and P(X=0) times P(Y=0) equals 1/4 times 1/4, which is 1/16, they are not independent.
To summarize, we found marginal distributions and verified independence through calculations. Well done!
Now, let's move to our second example, which involves a continuous joint probability distribution defined as f(x,y) = 4xy for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. What do you think we need to do first here?
First, we should check if it's a valid probability density function by integrating it over the defined range.
Correct! If we integrate f(x,y) over that area, what should we expect the result to be?
It should equal 1 if it's valid.
Yes! Let's perform the integration. What are the bounds?
From 0 to 1 for both x and y.
We'll integrate: ∫ from 0 to 1, ∫ from 0 to 1 of 4xy dy dx. Can anyone calculate that?
After calculating, it equals 1, so it’s valid!
Excellent! Now that we've verified it's a valid pdf, how do we find the marginal probability density functions for X and Y?
We integrate f(x,y) with respect to the other variable.
Exactly! So if we integrate f(x,y) with respect to y, what's the result for f_X(x)?
The result would be f_X(x) = 2x after integration!
Well done! How about the marginal for Y?
That would lead us to f_Y(y) = 2y after integration.
Right! Finally, how can we check if X and Y are independent?
We see if f(x,y) equals f_X(x) times f_Y(y), and if it does, they are independent.
Perfect! Since f(x,y) equals f_X(x) times f_Y(y), X and Y are independent as well. Great job, everyone!
A way to describe the relationship between two or more random variables.
The probability distribution of a single variable irrespective of others.
If two variables are independent, knowing the value of one provides no information about the other.
If PMF you must employ, marginal luck will you enjoy!
Once upon a time, two random friends, X and Y, met at a probability party. They learned that if they shared their info too closely, they couldn't be independent, revealing more about each other than they'd like!
Remember 'MIG' for: Marginal = Integrate for Joint distributions!
JIM for Joint Independence Marginal: Just check their compatibility!
Term
Joint PMF
Click to flip
Term
Marginal PDF
Click to flip
Term
Independence
Click to flip
A function that gives the probability that two discrete random variables equal specific values.
A function that describes the likelihood for continuous random variables to occur at a given point in the defined range.
The probability distribution of a subset of a collection of random variables.
Two random variables are independent if the joint probability or density function equals the product of their individual distributions.
This section presents example problems illustrating the use of joint probability distributions, covering both discrete and continuous cases.
Example Problems provide practical illustrations of joint probability distributions, showing how marginal distributions are calculated and how to check for independence in discrete and continuous cases. Two examples demonstrate the application of these concepts effectively.
In this section, we explore example problems related to joint probability distributions for both discrete and continuous cases. The first example details a discrete joint probability mass function defined for a limited set of outcomes, focusing on calculating marginal distributions and checking for independence. The second example covers a continuous joint probability density function, validating its total probability and similarly deriving marginal distributions while confirming independence between the random variables. These examples reinforce key concepts of joint distributions, such as marginalization and the criteria for independence, thus providing a practical context to the theoretical foundations discussed earlier in the chapter.