Question
More concepts and notation We use capital letters X and Y to name random variables, and we use lower case letters x and y for instances of their respective outcomes. These are drawn from particular sets A and B: x ∈ {a 1 , a 2 , ...a J }, and y ∈ {b 1 , b 2 , ...b K }. The probability of any particular outcome p(x = a i ) is denoted p i , for 0 ≤ p i ≤ 1 and with P i p i = 1. An ensemble is just a random variable X . A joint ensemble ‘XY ’ is an ensemble whose outcomes are ordered pairs x, y with x ∈ {a 1 , a 2 , ...a J } and y ∈ {b 1 , b 2 , ...b K }. The joint ensemble XY defines a probability distribution p(x, y ) over all the JK possible joint outcomes x, y. Marginal probability: From the Sum Rule, we can see that the probability of X taking on any particular value x = a i equals the sum of the joint probabilities of this outcome for X and all possible outcomes for Y : p(x = a i ) = X y p(x = a i , y). We usually simplify this notation for the marginal probabilities to: p(x) = X y p(x, y) and p(y ) = X x p(x, y)
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Question
More concepts and notation We use capital letters X and Y to name random variables, and we use lower case letters x and y for instances of their respective outcomes. These are drawn from particular sets A and B: x ∈ {a 1 , a 2 , ...a J }, and y ∈ {b 1 , b 2 , ...b K }. The probability of any particular outcome p(x = a i ) is denoted p i , for 0 ≤ p i ≤ 1 and with P i p i = 1. An ensemble is just a random variable X . A joint ensemble ‘XY ’ is an ensemble whose outcomes are ordered pairs x, y with x ∈ {a 1 , a 2 , ...a J } and y ∈ {b 1 , b 2 , ...b K }. The joint ensemble XY defines a probability distribution p(x, y ) over all the JK possible joint outcomes x, y. Marginal probability: From the Sum Rule, we can see that the probability of X taking on any particular value x = a i equals the sum of the joint probabilities of this outcome for X and all possible outcomes for Y : p(x = a i ) = X y p(x = a i , y). We usually simplify this notation for the marginal probabilities to: p(x) = X y p(x, y) and p(y ) = X x p(x, y)
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Question
More concepts and notation We use capital letters X and Y to name random variables, and we use lower case letters x and y for instances of their respective outcomes. These are drawn from particular sets A and B: x ∈ {a 1 , a 2 , ...a J }, and y ∈ {b 1 , b 2 , ...b K }. The probability of any particular outcome p(x = a i ) is denoted p i , for 0 ≤ p i ≤ 1 and with P i p i = 1. An ensemble is just a random variable X . A joint ensemble ‘XY ’ is an ensemble whose outcomes are ordered pairs x, y with x ∈ {a 1 , a 2 , ...a J } and y ∈ {b 1 , b 2 , ...b K }. The joint ensemble XY defines a probability distribution p(x, y ) over all the JK possible joint outcomes x, y. Marginal probability: From the Sum Rule, we can see that the probability of X taking on any particular value x = a i equals the sum of the joint probabilities of this outcome for X and all possible outcomes for Y : p(x = a i ) = X y p(x = a i , y). We usually simplify this notation for the marginal probabilities to: p(x) = X y p(x, y) and p(y ) = X x p(x, y)
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owner: iamc - (no access) - InfoTheoryNotes2020 (1).pdf, p21

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