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Questions and Answers
What does the variable X represent in the given example?
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)?
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)?
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?
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?
If P(X, Y) is the joint distribution, what does it represent?
If P(X, Y) is the joint distribution, what does it represent?
Which of the following probabilities is not part of the table for P(X, Y) as presented?
Which of the following probabilities is not part of the table for P(X, Y) as presented?
What does joint probability measure represent in the context of two discrete random variables X and Y?
What does joint probability measure represent in the context of two discrete random variables X and Y?
Which function calculates the marginal probability of X from the joint probability?
Which function calculates the marginal probability of X from the joint probability?
What is the relationship between joint probability and conditional probability?
What is the relationship between joint probability and conditional probability?
How can joint entropy of two random variables (X,Y) be expressed mathematically?
How can joint entropy of two random variables (X,Y) be expressed mathematically?
What does it mean if two discrete random variables X and Y are independent?
What does it mean if two discrete random variables X and Y are independent?
In calculating the marginal probability P(Y), which of the following equations correctly represents the summation over all outcomes in X?
In calculating the marginal probability P(Y), which of the following equations correctly represents the summation over all outcomes in X?
What is the significance of marginal probability functions in the study of joint probability?
What is the significance of marginal probability functions in the study of joint probability?
Which of the following correctly describes the calculation of P(X=1, Y=2) from the joint probability table?
Which of the following correctly describes the calculation of P(X=1, Y=2) from the joint probability table?
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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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