Question

5.18. A dart is equally likely to land at any point (X1, X2) inside a circular target of unit radius. Let Rand be the radius and angle of the point (X1, X2). (a)find the joint cdf of R and angle

Answer 1

The joint cumulative distribution function (CDF) of R (radius) and θ (angle) is given by:

F(R, θ) = (1/2) * (r^2) * θ / π

Since the dart is equally likely to land at any point inside a circular target of unit radius, the probability of the dart landing within a region A is proportional to the area of A on the unit circle.

This formula represents the probability of R being less than or equal to r and Θ being less than or equal to θ, considering a circular target of unit radius.

The expression holds for 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π. It is derived from the ratio of the area of the circular sector defined by R ≤ r and Θ ≤ θ to the total area of the unit circle, reflecting the equal likelihood of the dart landing at any point within the circular target.