Information Theory: Joint and Marginal Probability

Questions and Answers

What does the variable X represent in the given example?

The temperature in the town

The humidity levels

Whether it’s sunny or rainy (correct)

The area of the town

What is the significance of joint entropy H(X, Y)?

It calculates the total information of two independent variables.

It quantifies the uncertainty associated with two random variables. (correct)

It measures the uncertainty of one variable only.

It determines the average of two distinct measurements.

Using the provided probabilities, what is one component term in the calculation of H(X, Y)?

$rac{1}{6} log_{10} 6$

$rac{1}{18} log_{10} 18$ (correct)

$rac{1}{4} log_{10} 4$

$rac{1}{9} log_{10} 3$

What is the combined outcome of $rac{1}{6} log_{10} 18$, $rac{1}{9} log_{10} 9$, and $rac{1}{6} log_{10} 6$ in terms of joint entropy?

0.883 hartlys Signup and view all the answers

If P(X, Y) is the joint distribution, what does it represent?

The likelihood of both random variables occurring together Signup and view all the answers

Which of the following probabilities is not part of the table for P(X, Y) as presented?

$rac{2}{6}$ Signup and view all the answers

What does joint probability measure represent in the context of two discrete random variables X and Y?

The combination of the probabilities of both variables occurring together. Signup and view all the answers

Which function calculates the marginal probability of X from the joint probability?

P(X) = ∑ p(x,y) for y ∈ Y Signup and view all the answers

What is the relationship between joint probability and conditional probability?

Conditional probability represents the probability of one variable given another. Signup and view all the answers

How can joint entropy of two random variables (X,Y) be expressed mathematically?

H(X,Y) = - ∑ ∑ P(x,y) log(p(x,y)) Signup and view all the answers

What does it mean if two discrete random variables X and Y are independent?

Their joint probability equals the product of their marginal probabilities. Signup and view all the answers

In calculating the marginal probability P(Y), which of the following equations correctly represents the summation over all outcomes in X?

P(Y) = ∑ p(x,y) for x ∈ X Signup and view all the answers

What is the significance of marginal probability functions in the study of joint probability?

They summarize individual outcomes without considering the interaction. Signup and view all the answers

Which of the following correctly describes the calculation of P(X=1, Y=2) from the joint probability table?

P(X=1, Y=2) equals the specific entry in the joint probability matrix for X=1 and Y=2. Signup and view all the answers

Study Notes

Joint and Marginal Probability

Joint probability measures the likelihood of two events occurring together within a sample space Ω = X x Y.
It is defined as Pij = p(xi, yj) where X={x1,x2,…,xm} and Y={y1,y2,…,yn}.
Marginal probability functions are derived from joint probabilities:
- Pi(x) = Σ p(x, y) for all y in Y, gives the probability of x.
- P(y) = Σ p(x, y) for all x in X, gives the probability of y.
Conditional probability is calculated as p(xi | yj) = p(xi, yj) / p(yj).
Two random variables xi and yj are independent if p(xi, yj) = p(xi) * p(yj).

Example of Joint and Marginal Probability

Given a discrete joint distribution with probabilities across a table format.
Calculation for marginal probabilities:
- P(X=1) = 6/18 = 1/3, P(X=2) = 2/9, P(X=3) = 4/9.
- For Y: P(Y=1) = 1/3, P(Y=2) = 5/18, P(Y=3) = 7/18.
Joint probability P(X=1, Y=2) equals 1/9.

Joint Entropy

Joint entropy H(X, Y) quantifies the uncertainty associated with a pair of random variables (X,Y) based on their joint probability mass function.
Calculated as H(X, Y) = - Σ Σ P(x, y) log(p(x, y)), summing over all x in X and y in Y.
It reflects the unpredictability of the system as a whole, taking into account the interaction between the two variables.

Example of Joint Entropy Calculation

A scenario examines the weather conditions represented by random variables X (weather condition) and Y (temperature).
The resulting joint distribution influences the entropy calculation.
Example calculation yields H(X, Y) = 1.5 bits, demonstrating the information contained in the joint distribution of the two variables.

Additional Example of Joint Entropy

Calculating joint entropy based on another discrete set of random variables and their joint distribution.
The result of the calculation is given in hartleys, with specific values calculated from the joint probabilities.
Resulting joint entropy H(X, Y) = 0.883 hartleys displays the level of uncertainty regarding the joint occurrence of X and Y.

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Description

Explore the foundational concepts of joint and marginal probability in this quiz focused on Information Theory. Test your understanding of probability measures and functions as they relate to sets and events. Perfect for students looking to deepen their knowledge in statistical concepts.