Suppose X1, X2 are iid with a common standard normal...

Suppose $X_{1}, X_{2}$ are iid with a common standard normal distribution. Find the joint pdf of $Y_{1}=X_{1}^{2}+X_{2}^{2}$ and $Y_{2}=X_{2}$ and the marginal pdf of $Y_{1}$. Hint: Note that the space of $Y_{1}$ and $Y_{2}$ is given by $-\sqrt{y_{1}}<y_{2}<\sqrt{y_{1}}, 0<y_{1}<\infty$.

Key Concepts

Support of Random Variables

The support of a random variable refers to the set of values where its probability density function is non-zero. In transformations, it is important to correctly determine the new support in the transformed variable space, as this influences the limits of integration and the proper normalization of the pdf.

Marginalization

Marginalization involves integrating the joint probability density function over one or more of its variables to obtain the marginal density of the remaining variables. This process 'sums out' the influence of the other variables and provides the distribution of the variable of interest irrespective of the others.

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of simultaneous occurrences of two or more random variables. When new random variables are defined via transformations, the joint pdf is obtained by substituting the inverse transformation into the original joint pdf and then multiplying by the absolute value of the Jacobian determinant to account for the volume change in the variable transformation.

Jacobian Determinant

The Jacobian determinant is a key component in performing multivariate transformations. It is used to adjust the probability density when changing variables. Specifically, it quantifies how differential volume elements are scaled under the transformation, ensuring that the resulting joint density of the new variables integrates correctly to one over the new space.

Transformation of Random Variables

This concept involves finding the distribution of new random variables that are defined as functions of original random variables. When we perform such a transformation, we map outcomes from the original space to a new space and need to adjust the probability density accordingly. This is typically done using the change-of-variables technique, which ensures that probability is conserved through the transformation.

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