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The cdf of a two-dimensional Farlie-Gumbel-Morgenstern distribution is given by $$F(x,y)=\underbrace{(1-e^{\alpha_1x+\beta_1x^2})}_{F_x(x)}\underbrace{(1-e^{\alpha_1x+\beta_1x^2})}_{F_Y(y)}[1+\lambda\,e^{\alpha_1x+\beta_1x^2}\,e^{\alpha_1x+\beta_1x^2}]$$ with $|\lambda|<1$ and $\alpha_i,\beta_i>0$ for $i=1,2$. This implies that, if $(X,Y)\sim F$, then $...


Just consider the conditional distribution of $Y|Z$. This can be characterized as $P(Y = z|Z = z) = 0.5$ and $P(Y = -z|Z = z) = 0.5$. This is no normal distribution and therefore the distribution of $(Y, Z)$ is not bivariate normal (because for a multivariate normal distribution the conditional distributions are also normal).

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