6.22. The joint density function of X and Y is f(x, y) = { x + y 0 < x < 1, 0 < y < 1 0 otherwise (a) Are X and Y independent? (b) Find the density function of X. (c) Find P{X + Y < 1}.
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Step 1: To determine if X and Y are independent, we need to calculate the marginal density functions of X and Y and then check if the joint density function can be expressed as the product of the marginal density functions. Show more…
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The joint density function of the random variables $X$ and $Y$ is $$ f(x, y)=\left\{\begin{array}{ll} 6 x, & 0<x<1,0<y<1-x, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Show that $X$ and $Y$ are not independent. (b) Find $P(X>0.3 \mid Y=0.5)$.
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