Question

What specific relation must exist between a and b, in order for f x y ( , ) to indeed be a legitimate joint pdf of two continuous random variables x and y? show all work

Answer 1

The answer will be a = 1.6 and b = 11 /12

What is integration?

Integration is defined as the summing up of the small parts to find the total value.

The solution of the given question is as follows:-

The joint pdf by x and y

1 < x <2

0 < x <1

In point a:

$K\int^1_{y=0} \int_{x=1}^2( 2y-\dfrac{x}{4}dxdy=1\\\\\\K\int _{y=0}^12y-\dfrac{x^2}{8}\int_1^2dy=1\\\\\\K\ y^2\int_{y=0}^1-\dfrac{3}{8}=1\\\\\\K(1-\dfrac{3}{8})=1\\\\\\K=\dfrac{8}{5}$

K = 1.6

In point b:

In the original pdf of x

$f(x)=K\int_{y=0}^1(2y-\dfrac{x}{4})dy=k\\\\\\K\int_1^2(x-\dfrac{x^2}{4})dx=K\\\\\\K(\dfrac{x^2}{2}-\dfrac{x^3}{12})_1^2=K\\\\\\K(\dfrac{1}{2}\times 3)-\dfrac{1}{12}\times 7)=K\\\\\dfrac{3}{2}-\dfrac{7}{12}\\\\\\=\dfrac{11}{12}$

Therefore the answer will be a = 1.6 and b = 11 /12

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