Linked Questions
14 questions linked to/from Is joint normality a necessary condition for the sum of normal random variables to be normal?
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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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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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$X_i, X_j$ 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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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 $C$:
$$...
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Get joint distribution from pairwise marginal distribution
Assume we have 3 random variables $X_1,X_2,X_3$, and we know the pairwise marginal distribution $P(X_1,X_2), P(X_2,X_3), P(X_3,X_1)$, but we don't know anything else (such as conditional independence)....
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Marginal independence does not imply joint independence
Consider the random variables $X,Y,Z$.
Assume
$$
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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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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1
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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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2
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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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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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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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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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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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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?
