Question

For a tworandp m variables X∝Y, the joint denaity function is given as

fx,y(x,y)= 4/9xy,0 ≤ x ≤1,0 ≤ y ≤ 3.
Show that random vaviables x ∝ y, are stafistically independent.

Answer 1

The question asks to prove that the random variables X and Y are statistically independent given the joint density function fx,y(x,y) = 4/9xy. The proof involves finding the marginal density functions for X and Y and showing that their product equals the joint density function. Since the product of the marginal density functions equals the joint density function, X and Y are statistically independent.

Step-by-step explanation:

To show that random variables X and Y are statistically independent, we need to prove that the joint density function can be expressed as the product of the marginal density functions of X and Y separately. The joint density function is given by fx,y(x,y) = 4/9xy, where 0 ≤ x ≤ 1, and 0 ≤ y ≤ 3.

First, let's find the marginal density functions of X and Y. The marginal density function of X, fx(x), is obtained by integrating the joint density function over the entire range of Y:

fx(x) = ∫ 4/9xy dy from y=0 to y=3 = 4/9x(∫ y dy) = 4/9x[y^2/2]_0^3 = 4/9x(9/2) = 2x

Similarly, the marginal density function of Y, fy(y), is obtained by integrating the joint density function over the entire range of X:

fy(y) = ∫ 4/9xy dx from x=0 to x=1 = 4/9y(∫ x dx) = 4/9y[x^2/2]_0^1 = 4/9y(1/2) = 2y/9

Now that we have both marginal density functions fx(x) = 2x and fy(y) = 2y/9, we can find their product:

fx(x) * fy(y) = (2x) * (2y/9) = 4/9xy

Since fx(x) * fy(y) = fx,y(x,y), this shows that the joint density function is indeed the product of the two marginal density functions, which means X and Y are statistically independent.