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Let101
Let101
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We have a random variable $X\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$$\mathbf{X}\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Let $\mathbf{x}$ be the value taken by $X$$\mathbf{X}$. We want to bet on the event $x_1\ge x_2$, where $x_1$ and $x_2$ are respectively the first and second component of $\mathbf{x}$.

Question: What are sufficient and necessary conditions for $\mathbf{\Sigma}$ that guarantee $x_1\ge x_2$ with a desired probability $p\in\left(\tfrac12,1\right)$?

We have a random variable $X\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Let $\mathbf{x}$ be the value taken by $X$. We want to bet on the event $x_1\ge x_2$, where $x_1$ and $x_2$ are respectively the first and second component of $\mathbf{x}$.

Question: What are sufficient and necessary conditions for $\mathbf{\Sigma}$ that guarantee $x_1\ge x_2$ with a desired probability $p\in\left(\tfrac12,1\right)$?

We have a random variable $\mathbf{X}\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Let $\mathbf{x}$ be the value taken by $\mathbf{X}$. We want to bet on the event $x_1\ge x_2$, where $x_1$ and $x_2$ are respectively the first and second component of $\mathbf{x}$.

Question: What are sufficient and necessary conditions for $\mathbf{\Sigma}$ that guarantee $x_1\ge x_2$ with a desired probability $p\in\left(\tfrac12,1\right)$?

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Let101
Let101
  • 83
  • 6

Probabilistic problem on the covariance matrix of a multivariate normal distribution

We have a random variable $X\sim\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$, where $\mathcal{N}_d(\mathbf{\mu},\mathbf{\Sigma})$ is a $d$-dimensional multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Let $\mathbf{x}$ be the value taken by $X$. We want to bet on the event $x_1\ge x_2$, where $x_1$ and $x_2$ are respectively the first and second component of $\mathbf{x}$.

Question: What are sufficient and necessary conditions for $\mathbf{\Sigma}$ that guarantee $x_1\ge x_2$ with a desired probability $p\in\left(\tfrac12,1\right)$?