Question

A gun is aimed at a certain point, say, the origin of the co-ordinate system. Due to the random factors, the actual hit point can be any point (X,Y) in a circle of radius a with center at the origin. If the joint density function of (X,Y) can be assumed as f(x,y) = k, for r² + y2 so? and = 0, otherwise, Find the value of k

Answer 1

The value of k in the given joint density function f(x, y) = k is found by integrating over the area of the circle and setting the total probability to 1. The value of k is 1 / (πa^2), ensuring that the total probability over the circle is 1.

Step-by-step explanation:

The student asks for help in finding the value of k, where the joint density function of (X,Y) is given by f(x,y) = k within a circle of radius centered at the origin of a coordinate system. To find k, we need to ensure the total probability over the entire space is 1, as it's a fundamental property of probability density functions.

To do this, we integrate the density function over the area of the circle. The integral of a constant k over the area of a circle with radius a is equal to k times the area of the circle, which is πa^2. Setting this integral equal to 1 gives us:

∫∫ f(x, y) dA = k × (πa^2) = 1

Therefore, k = 1 / (πa^2).