The joint density function f(x,y)=53(xy+y^2) is defined for 0≤x≤2 and 0≤y≤1, with the support being the rectangular region between these bounds. It is shown to integrate to 1 over this region, confirming it is a valid joint density function. The marginal densities of X and Y are then calculated by integrating f(x,y) over the other variable. The probabilities P(X<Y) and expectation E(XY) are evaluated from the joint density. Finally, since f(x,y) does not factor into separate functions of x and y, X and Y are determined to be dependent random variables.
Introduction to ArtificiaI Intelligence in Higher Education
Joint Density Function Support, Marginals, Independence Check
1. Suppose X,Y have joint density function fX,Y(x,y)={53(xy+y2),0,0x2otherwise.and0y1 (a) Sketch
the support of joint distribution (X,Y). (b) Check that f is a genuine joint density function. (c) Find
the marginal density functions of X and Y. (d) Calculate the probability P(X<Y). (e) Calculate the
expectation E[XY]. (f) Determine whether X and Y are independent.
![Suppose X,Y have joint density function fX,Y(x,y)={53(xy+y2),0,0x2otherwise.and0y1 (a) Sketch
the support of joint distribution (X,Y). (b) Check that f is a genuine joint density function. (c) Find
the marginal density functions of X and Y. (d) Calculate the probability P(X<Y). (e) Calculate the
expectation E[XY]. (f) Determine whether X and Y are independent.](https://image.slidesharecdn.com/supposexyhavejointdensityfunctionfxyxy53xyy2-230321044057-10180d1d/85/suppose-xy-have-joint-density-function-fxyxy53xyy2pdf-1-320.jpg)
