Question
The joint density of $X$ and $Y$ is$$f(x, y)=c\left(x^{2}-y^{2}\right) e^{-x} \quad 0 \leq x<\infty,-x \leq y \leq x$$Finel the conditional distribution of $Y$, given $X=x$.
Step 1
The joint density function must integrate to 1 over the entire domain. Therefore, we have: $$ \begin{aligned} 1 &= \int_{0}^{\infty} \int_{-x}^{x} c(x^{2}-y^{2}) e^{-x} dy dx \\ &= c \int_{0}^{\infty} \left[ \int_{-x}^{x} (x^{2}-y^{2}) dy \right] e^{-x} dx \\ &= Show more…
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Key Concepts
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Problems
The joint probability density function of $X$ and $Y$ is given by $$ f(x, y)=c\left(y^{2} \div x^{2}\right) e^{-y} \quad-y \leq x \leq y, 0<y<\infty $$ (a) Find $c$. (b) Find the marginal densities of $X$ and $Y$.
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