diff --git a/05-Multivariate-Gaussians.ipynb b/05-Multivariate-Gaussians.ipynb index 0518297..43d8afb 100644 --- a/05-Multivariate-Gaussians.ipynb +++ b/05-Multivariate-Gaussians.ipynb @@ -727,7 +727,7 @@ "source": [ "This is a plot of multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}10&0\\\\0&4\\end{smallmatrix}]$. The three dimensional shape shows the probability density for any value of $(X, Y)$ in the z-axis. I have projected the variance for x and y onto the walls of the chart - you can see that they take on the Gaussian bell curve shape. The curve for $X$ is wider than the curve for $Y$, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. The highest point of the 3D surface is at the the means for $X$ and $Y$. \n", "\n", - "All multivariate Gaussians have this shape. If we think of this as the Gaussian for the position of a dog, the z-value at each point of ($X, Y$) is the probability density of the dog being at that position. Strictly speaking this is the *joint probability density function*, which I will define soon. So, the dog has the highest probability of being near (2, 17), a modest probability of being near (5, 14), and a very low probability of being near (10, 10). As with the univariate case this is a *probability density*, not a *probability*. Continuous distributions have an infinite range, and so the probability of being exactly at (2, 17), or any other point, is 0%. We can compute the probability of being within a given range by computing the area under the curve with an integral." + "All multivariate Gaussians have this shape. If we think of this as the Gaussian for the position of a dog, the z-value at each point of ($X, Y$) is the probability density of the dog being at that position. Strictly speaking this is the *joint probability density function*, which I will define soon. So, the dog has the highest probability of being near (2, 17), a modest probability of being near (5, 14), and a very low probability of being near (10, 10). As with the univariate case this is a *probability density*, not a *probability*. Continuous distributions have an infinite range, and so the probability of being exactly at (2, 17), or any other point, is 0%. We can compute the probability of being within a given range by computing the volume under the surface with an integral." ] }, {