Linked Questions

1 vote
0 answers
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?
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  • 924
8 votes
2 answers
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 ...
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  • 97
7 votes
3 answers
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 ...
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  • 3,227
7 votes
2 answers
7k views

Covariance matrix for a linear combination of correlated Gaussian random variables

Supposing $X$ and $Y$ are random variables with a joint bivariate normal distribution and covariance matrix $\Sigma_{XY}$. Consider the following linear combination for constants $A$, $B$ and $C$: $$...
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  • 995
10 votes
1 answer
1k views

Get joint distribution from pairwise marginal distribution

Assume we have 3 random variables $X_1,X_2,X_3$, and we know the pairwise marginal distribution $P(X_1,X_2), P(X_2,X_3), P(X_3,X_1)$, but we don't know anything else (such as conditional independence)....
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6 votes
1 answer
2k views

Marginal independence does not imply joint independence

Consider the random variables $X,Y,Z$. Assume $$ X\perp Z $$ $$ Y\perp Z $$ $$ X\perp Y $$ Can we say $$ (X,Y)\perp Z $$ My intuition is that we can't because $$ p_{X,Y}=p_X\times p_Y=p_{X|Z} \...
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  • 421
1 vote
1 answer
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? ...
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  • 984
2 votes
1 answer
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 ...
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1 vote
2 answers
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 ...
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  • 21
2 votes
1 answer
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 ...
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3 votes
1 answer
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 ...
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2 votes
0 answers
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 ...
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0 votes
0 answers
22 views

Multivariate normal distribution property [duplicate]

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 ...
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0 votes
0 answers
17 views

Gaussian Distribution [duplicate]

Assume we have two continuous Normal RV "X" and "Y". how can I show the conditional PDF f(X|Y) and f(Y|X) is Normal?
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