Skip to main content
Cross Validated

Return to Question

Tweeted twitter.com/StackStats/status/898291323681112064
added LaTeX
Source Link
jld
jld
  • 20.8k
  • 2
  • 64
  • 69

Let (X,Y)$(X,Y)$ be a 2D gaussianGaussian, with non-zero correlation. Let I$I$ be an interval centered around the mean of X$X$, and J$J$ an interval centered around the mean of Y$Y$. I want to show

$$ P(X \in I | Y \in J) \geq P(X \in I), $$ i.e. that P(X is in I | Y is in J)the conditional probability is superiorgreater than or equal than P(X is in I) without conditionto the unconditional one.

I am sure this is true thanks to simulation, but I cannot find a proof of this result. It feels very natural that, for two correlated gaussians, knowing that one is closer to its mean implies that the probability that the other is also closer to its mean is higher. I tried using the conditional density of X | Y=a$X | Y=a$ and integrating over a$a$, but I couldn't find a minoration.

Let (X,Y) be a 2D gaussian, with non-zero correlation. Let I be an interval centered around the mean of X and J an interval centered around the mean of Y. I want to show that P(X is in I | Y is in J) is superior or equal than P(X is in I) without condition.

I am sure this is true thanks to simulation, but I cannot find a proof of this result. It feels very natural that, for two correlated gaussians, knowing that one is closer to its mean implies that the probability that the other is also closer to its mean is higher. I tried using the conditional density of X | Y=a and integrating over a, but I couldn't find a minoration.

Let $(X,Y)$ be a 2D Gaussian, with non-zero correlation. Let $I$ be an interval centered around the mean of $X$, and $J$ an interval centered around the mean of $Y$. I want to show

$$ P(X \in I | Y \in J) \geq P(X \in I), $$ i.e. that the conditional probability is greater than or equal to the unconditional one.

I am sure this is true thanks to simulation, but I cannot find a proof of this result. It feels very natural that, for two correlated gaussians, knowing that one is closer to its mean implies that the probability that the other is also closer to its mean is higher. I tried using the conditional density of $X | Y=a$ and integrating over $a$, but I couldn't find a minoration.

Source Link
Romain Nijeul
Romain Nijeul
  • 51
  • 1

Conditional probability of correlated gaussians

Let (X,Y) be a 2D gaussian, with non-zero correlation. Let I be an interval centered around the mean of X and J an interval centered around the mean of Y. I want to show that P(X is in I | Y is in J) is superior or equal than P(X is in I) without condition.

I am sure this is true thanks to simulation, but I cannot find a proof of this result. It feels very natural that, for two correlated gaussians, knowing that one is closer to its mean implies that the probability that the other is also closer to its mean is higher. I tried using the conditional density of X | Y=a and integrating over a, but I couldn't find a minoration.