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Let A = {1, 2, ..., n}$A = \{1, 2,\dots, n\}$, and let X$X$ and Y$Y$ be two random variables on the same space, taking values in A$A$, and distributions given by: P(X = i) = $p_i$$P(X = i) = p_i$, and P(Y = i) = $q_i$$P(Y = i) = q_i$, for any i in {1,2,..$i\in\{1,2,\dots,n\}$.,n}

What is the maximum possible value of P(X = Y)$P(X = Y)$?

Basically, we have to choose a good joint distribution of X$X$ and Y$Y$, such that the sum of diagonal entries in the joint distribution table/matrix is maximized. For n = 2$n = 2$, this is easy to do, and I get the answer (1 - |$p_1$ - $q_1$|)$(1 - |p_1 - q_1|)$. But even for n = 3$n = 3$, I am not able to do much. A solution/strategy for general n would be appreciated.

Let A = {1, 2, ..., n}, and let X and Y be two random variables on the same space, taking values in A, and distributions given by: P(X = i) = $p_i$, and P(Y = i) = $q_i$, for any i in {1,2,...,n}

What is the maximum possible value of P(X = Y)?

Basically, we have to choose a good joint distribution of X and Y, such that the sum of diagonal entries in the joint distribution table/matrix is maximized. For n = 2, this is easy to do, and I get the answer (1 - |$p_1$ - $q_1$|). But even for n = 3, I am not able to do much. A solution/strategy for general n would be appreciated.

Let $A = \{1, 2,\dots, n\}$, and let $X$ and $Y$ be two random variables on the same space, taking values in $A$, and distributions given by: $P(X = i) = p_i$, and $P(Y = i) = q_i$, for any $i\in\{1,2,\dots,n\}$.

What is the maximum possible value of $P(X = Y)$?

Basically, we have to choose a good joint distribution of $X$ and $Y$, such that the sum of diagonal entries in the joint distribution table/matrix is maximized. For $n = 2$, this is easy to do, and I get the answer $(1 - |p_1 - q_1|)$. But even for $n = 3$, I am not able to do much. A solution/strategy for general n would be appreciated.

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How to choose two random variables taking values in a finite space, with given distributions, such that probability that they are equal is maximized?

Let A = {1, 2, ..., n}, and let X and Y be two random variables on the same space, taking values in A, and distributions given by: P(X = i) = $p_i$, and P(Y = i) = $q_i$, for any i in {1,2,...,n}

What is the maximum possible value of P(X = Y)?

Basically, we have to choose a good joint distribution of X and Y, such that the sum of diagonal entries in the joint distribution table/matrix is maximized. For n = 2, this is easy to do, and I get the answer (1 - |$p_1$ - $q_1$|). But even for n = 3, I am not able to do much. A solution/strategy for general n would be appreciated.