1 vote
110 views

### Conditions for gaussianity of sum of Gaussians [duplicate]

Under which conditions is the sum of a finite number of Gaussian random variables a Gaussian?
22k views

### If $X$ and $Y$ are normally distributed random variables, what kind of distribution their sum follows?

I was reading this question. It is about notation but I would like to ask something regarding the sum of two normally distributed random variables. If $X$ is a normally distributed random variable ...
961 views

### $X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?

I've seen the statement: It's possible that random variables $X_i, X_j$ are independent for $i≠j$, but $X_1, X_2, X_3$ are dependent. I haven't been able to find examples of this though. Any ...
7k views

1 vote
1k views

### Can the sum of two non-normal be normal?

I know that the sum of two normally distributed random variables is also normal. But what about the opposite? Can the sum (or substraction) of two non-normally distributed random variables be normal? ...
789 views

### How to check if two random Gaussian vectors are jointly Gaussian?

Given the following system of equations: $\textbf{y} = \textbf{x}_{1} + \sum_{l=2}^{L}\textbf{x}_{l} + \textbf{w}$ where $\textbf{y}$, $\textbf{x}_{l}, \; \forall l$ and $\textbf{w}$ are Gaussian ...
1 vote
560 views

### Dependent / Independent random variables with identical Cumulative distribution function

I'm stuck with an assignment, hope you guys can help. Question: Show, that there exist random variables $X,Y,X',Y'$ on a Probability Space $(\Omega, \mathscr{F},P)$, so that $X$ and $Y$ are not ...
367 views

### Two random variables not correlated with normal distribution, whose sum is normal but which are not independent

The example based on the Rademacher distribution of two random variables, $X$ and $Y$, normally standard distributed, not correlated, but not independent, is well known. Additionally, in this example ...
130 views

### Normality of sum of normal random variables

If $(X,Y)$ and $(X+Y,Z)$ both follow nondegenerate bivariate Gaussian distributions, is it possible that $(X,Y,Z)$ follow a nondegenerate trivariate distribution that is not Gaussian? I want to make a ...
60 views

### Necessary condition for sum of normals to be a normal

What is the necessary condition for a sum $Z=X+Y$ of two normal random variables $X$ and $Y$ to be a normal random variable? If this is too difficult to state in general, what are some sufficient ...
The Econonometrics book by Hansen mentions this property of a multivariate normal distribution: If $X∼N(\mu,\Sigma)$ then $AX+b∼N(Aμ+b,AΣA^\prime)$. It goes to say that a simple implication of this ...