Question

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below.

f(x, y) = { 1/πR² x² + y² < R²
{0 , otherwise

what is the probability that the selected point is within R/3 of the center of the circular region?

Answer 1

The probability that the selected point is within R/3 of the center of the circular region is 1/9.

Step-by-step explanation:

To find the probability that the selected point is within R/3 of the center of the circular region, we need to determine the area of the circular region within R/3 of the center and divide it by the total area of the circle.

The radius of the smaller circle is R/3. The area of the smaller circle is given by A = π(R/3)^2 = πR^2/9. The area of the larger circle is given by A = πR^2. Therefore, the probability is equal to the ratio of the areas, P = (πR^2/9)/(πR^2) = 1/9.

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An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected.