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If two random variable (independent) is given say $X$ which is real and distributed $~N(0,1)$ and a discrete random variable $\alpha$ that takes +1 or -1 with probability half each. A transformation is given as $Y=\alpha X$. I need to find the joint distribution between $X$ and $Y$.

Is it through the bivariate transformation considering one tranformation as $U=\alpha X$ and another one as V=X and I can find their joint distribution or any easy intuitive way to figure out this?

I am following this : https://onlinecourses.science.psu.edu/stat414/node/129

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\begin{align} P(X \leq x, Y \leq y) &=P(\alpha=1)P(X \leq x, \alpha X \leq y| \alpha =1)+ P(\alpha=-1)P(X \leq x, \alpha X \leq y| \alpha =-1)\\ &=\frac12 P(X \leq x, X \leq y)+ \frac12P(X \leq x, -X \leq y) \\ &=\frac12 P(X \leq \min(x,y))+ \frac12 P(-y \leq X \leq x) \end{align}

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    thanks!, by the way y will be always less than or equal o x, right? so min(x,y)=y, isnt it and the joint distribution is Gaussian? – 2017-01-11
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    $Y \leq X$ but $x$ and $y$ are numbers that we want to evaluate the cdf, so $y$ need not be smaller than $x$. – 2017-01-11
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    Thanks understood, Sorry to ask you, can you suggest a good reference on probabiity random variable, I am a beginner in this course. – 2017-01-11
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    I used to browsed through Sheldon Ross "A first course in probability" , Bersekas "Introduction to probability", notes by David Gamarnik (MIT OCW fundamentals of probability). – 2017-01-11