Question

A dairy farm produces two different types of milk. Let X= the amount (in liter) of type A on hand and Y= the amount of type B on hand. Suppose the joint PDF of X and Y is given by f

X,Y
​
(x,y)={
kxy
0
​

x⩾0,y⩾0,20⩽x+y⩽30
otherwise
​
where k is a certain constant to be determined. 1. Draw the region of positive density, and determine the value of k.

Answer 1

The question regards a joint probability density function for two continuous variables, where you must find the constant 'k' and draw the positive density region. The region is a triangular area on the xy-plane, constrained by the bounds given. The constant 'k' is found by integrating the function over the region and setting the total area to 1.

Step-by-step explanation:

The question involves finding the constant k for the joint probability density function (pdf) of two continuous random variables X and Y. To find this constant, we need to ensure that the total area under the pdf equals 1 because this represents the entire probability space.

To draw the region of positive density, we first identify the constraints given by 20 ≤ x + y ≤ 30 and x ≥0, y ≥0. This region is a triangle on the xy-plane bounded by the lines x + y = 20, x + y = 30, and the axes. We need to solve the integral of kxy over this region to find k:

∫∫_{x=0}^{10}∫_{y=20-x}^{30-x} kxy dydx = 1

After evaluating the double integral, we will find the value of k that normalizes the pdf.