Let the random variables X1 and X2 have the joint pdf f(x1, x2)=1 / π, for (x1-1)^2+(x2+2)^2<1,

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Let the random variables X1 and X2 have the joint pdf f(x1,...

Let the random variables $X_{1}$ and $X_{2}$ have the joint pdf $f\left(x_{1}, x_{2}\right)=1 / \pi$, for $\left(x_{1}-1\right)^{2}+\left(x_{2}+2\right)^{2}<1$, zero elsewhere. Find $f_{1}\left(x_{1}\right)$ and $f_{2}\left(x_{2}\right)$. Are $X_{1}$ and $X_{2}$ independent?

Key Concepts

Joint Probability Density Function

A joint probability density function (pdf) specifies the likelihood of two or more random variables occurring simultaneously. It is defined over a region in the multidimensional space and must satisfy the property that its integral over the entire space equals one. In problems like this one, the joint pdf is used to describe the combined behavior of the random variables over a specific region.

Marginal Probability Density Function

A marginal probability density function is obtained by integrating the joint pdf with respect to one or more of the variables, thus ‘summing out’ the effect of those variables. This provides the probability distribution for a single variable, independent of the others.

Independence of Random Variables

Two random variables are independent if and only if their joint pdf factors into the product of their marginal pdfs for all values in their domain. This implies that the occurrence of one variable does not affect the probability distribution of the other. When the joint pdf is not factorable in this manner, the variables are dependent.

Uniform Distribution on a Region

A uniform distribution on a region is characterized by a constant probability density over the support of the distribution and zero elsewhere. This implies that every measurable subset of the region of equal size has the same probability, which is a key feature in problems involving uniform probability over geometric shapes such as circles or rectangles.

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