Question

Let (y1, y2) denote the coordinates of a point chosen at random inside a unit circle whose center is at the origin. that is, y1 and y2 have a joint density function given by f(y1, y2) = 1 , y12 y22 ≤ 1, 0, elsewhere. find p(y2 ≤ y1).

Answer 1

To find the probability that y2 is less than or equal to y1, we need to calculate the area under the density curve where y2 is less than or equal to y1.

Step-by-step explanation:

To find p(y2 ≤ y1), we need to calculate the probability that y2 is less than or equal to y1.

Given that the coordinates of a point chosen at random inside a unit circle have a joint density function of f(y1, y2) = 1, we can determine the probability by finding the area under the density curve.

The area between two points, a and b, corresponds to the probability that the variable falls between those two values, i.e., P(a ≤ x ≤ b). In this case, we need to find P(y2 ≤ y1) which is the area under the curve where y2 is less than or equal to y1.

Since the density function is constant with respect to the area, the area under the curve where y2 ≤ y1 is the same as the area of a unit square.