# Is sum of Gaussians Gaussian?

Table of Contents

## Is sum of Gaussians Gaussian?

The sum of two Gaussian variables is Gaussian. random variables means that the mean of the resultant Gaussian will be the sum of the input means and the variance of the sum will be the sum of the input variances.

## What is the sum of normal distribution?

This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).

## What is the sum under a normal distribution curve?

The sum of two normally distributed random variables is normal if the two random variables are independent or if the two random variables are jointly normally distributed. In the latter case the variance of the sum is the sum of the variances plus 2 times the covariance.

## How do you normalize a Gaussian distribution?

The Gaussian distribution arises in many contexts and is widely used for modeling continuous random variables. p(x | µ, σ2) = N(x; µ, σ2) = 1 Z exp ( − (x − µ)2 2σ2 ) . The normalization constant Z is Z = √ 2πσ2.

## What is the sum of random variables?

For any two random variables X and Y, the expected value of the sum of those variables will be equal to the sum of their expected values. The proof, for both the discrete and continuous cases, is rather straightforward.

## How do you compare two normal distributions?

The simplest way to compare two distributions is via the Z-test. The error in the mean is calculated by dividing the dispersion by the square root of the number of data points….So far this example:

- X1 = 51.5.
- X2 = 39.5.
- X1 – X2 = 12.
- σx1 = 1.6.
- σx2 = 1.4.
- sqrt of σx12 + σx22 =sqrt(1.62 + 1.42) = sqrt(2.56 +1.96) = 2.1.

## What is the sum of two independent random variables?

For two random variables X and Y, the additivity property E(X+Y)=E(X)+E(Y) is true regardless of the dependence or independence of X and Y. But variance doesn’t behave quite like this. Let’s look at an example.

## Why normal curve is bell shaped?

The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, so the right side of the center is a mirror image of the left side. The normal distribution is often called the bell curve because the graph of its probability density looks like a bell.

## How do we standardize a normal distribution?

Any normal distribution can be standardized by converting its values into z-scores….Standardizing a normal distribution

- A positive z-score means that your x-value is greater than the mean.
- A negative z-score means that your x-value is less than the mean.
- A z-score of zero means that your x-value is equal to the mean.

## How do you normalize different distributions?

Three obvious approaches are:

- Standardizing the variables (subtract mean and divide by stddev ).
- Re-scaling variables to the range [0,1] by subtracting min(variable) and dividing by max(variable) .
- Equalize the means by dividing each value by mean(variable) .

## Which is a weighted sum of Gaussian distributions?

Well, we start by defining the probability model. The probability of observing any observation, that is the probability density, is a weighted sum of K Gaussian distributions (as pictured in the previous section) : Each point is a mixture of K weighted Gaussians which are parameterized by a mean and a covariance matrix.

## How is a mixture of Gaussians parameterized?

Each point is a mixture of K weighted Gaussians which are parameterized by a mean and a covariance matrix. So overall, we can describe the probability of observing a specific observation as a mixture.

## What is the maximum likelihood for a Gaussian mixture?

By a similar argument: Maximum Likelihood for Gaussian Mixture Models Plan of Attack: 1. ML for a single Gaussian 2. ML for a fully-observed mixture 3. ML for a hidden mixture Maximum Likelihood for Fully-Observed Mixture ● “Observed Mixture” means we receive datapoints (x,α).

## How to know where the Gaussians are to be located?

Intuitively, for one selected observation and one selected Gaussian, the probability of the observation belonging to the cluster would be the ratio between the Gaussian value and the sum of all the Gaussians. Something like: Ok, but how do know where the Gaussians are to be located in the above plot (ie. how do we find the Gaussian parameters)?