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

After going through each question, the final answers come out to be as: (a) Joint CDF of R and Θ:
`\(F_(R,\Theta)(r, \theta) = r^2\)`. (b) Marginal CDFs of R and Θ:
`\(F_R(r) = (1)/(3)r^3, \quad F_\Theta(\theta) = \theta^2\)`. (c) Joint PDF of R and Θ:
`\(f_(R,\Theta)(r, \theta) = 2r\)`. (d) Marginal PDFs of R and Θ:
`\(f_R(r) = r^2, \quad f_\Theta(\theta) = 2\theta\)`. (e) Probability in the First Quadrant with Radius > 0.5:
`\(P(\text{First Quadrant and } R > 0.5) = 0.25\)`.

Let's go through each part step by step:

(a) Joint CDF of R and Θ:

The joint cumulative distribution function (CDF) is given by:

`\[ F_(R,\Theta)(r, \theta) = P(R \leq r, \Theta \leq \theta) \]`

For a circular target of unit radius, we can express this as:

`\[ F_(R,\Theta)(r, \theta) = P(√(X_1^2 + X_2^2) \leq r, \text{ atan2}(X_2, X_1) \leq \theta) \]`

Since
`\(X_1\)` and
`\(X_2\)` are equally likely to land anywhere in the circle, we can express this in terms of the area of the circle:

`\[ F_(R,\Theta)(r, \theta) = \frac{\text{Area of the circle within radius } r \text{ and angle } \theta}{\text{Total area of the circle}} \]`

The area of the circle within radius r is
`\( \pi r^2 \)`, and the total area of the circle is
`\( \pi \)` (since it's a unit radius circle).

`\[ F_(R,\Theta)(r, \theta) = (\pi r^2)/(\pi) = r^2 \]`

So, the joint CDF is
`\( F_(R,\Theta)(r, \theta) = r^2 \)`.

(b) Marginal CDFs of R and Θ:

To find the marginal CDFs, we integrate the joint CDF over the other variable:

`\[ F_R(r) = P(R \leq r) = \int_(-\infty)^(\infty) F_(R,\Theta)(r, \theta) \, d\theta \]\[ F_\Theta(\theta) = P(\Theta \leq \theta) = \int_(0)^(r) F_(R,\Theta)(r, \theta) \, dr \]`

Since we've already found
`\( F_(R,\Theta)(r, \theta) = r^2 \)`, we can substitute this into the above equations:

`\[ F_R(r) = \int_(0)^(r) r^2 \, dr = (1)/(3)r^3 \]\[ F_\Theta(\theta) = \theta^2 \]`

So, the marginal CDFs are
`\( F_R(r) = (1)/(3)r^3 \)` and
`\( F_\Theta(\theta) = \theta^2 \)`.

(c) Joint PDF of R and Θ:

The joint probability density function (PDF) is the derivative of the joint CDF:

`\[ f_(R,\Theta)(r, \theta) = (\partial^2)/(\partial r \partial \theta) F_(R,\Theta)(r, \theta) \]`

Since
`\( F_(R,\Theta)(r, \theta) = r^2 \)`, the joint PDF is:

`\[ f_(R,\Theta)(r, \theta) = (\partial^2)/(\partial r \partial \theta) (r^2) = 2r \]`

(d) Marginal PDFs of R and Θ:

To find the marginal PDFs, we take the partial derivatives of the marginal CDFs:

`\[ f_R(r) = (\partial)/(\partial r) F_R(r) = r^2 \]\[ f_\Theta(\theta) = (\partial)/(\partial \theta) F_\Theta(\theta) = 2\theta \]`

So, the marginal PDFs are
`\( f_R(r) = r^2 \)` and
`\( f_\Theta(\theta) = 2\theta \)`.

(e) Probability in the First Quadrant with Radius > 0.5:

The probability that the point is in the first quadrant
`(\(0 \leq \Theta \leq (\pi)/(2)\))` and that the radius is greater than 0.5 is given by:

`\[ P(\text{First Quadrant and } R > 0.5) = F_(R,\Theta)(0.5, (\pi)/(2)) \]`

Substitute the values into the joint CDF:

`\[ P(\text{First Quadrant and } R > 0.5) = (0.5)^2 = 0.25 \]`

So, the probability is
`\( P(\text{First Quadrant and } R > 0.5) = 0.25 \)`.

The complete question is:
A dart is equally likely to land at any point inside a circular target of unit radius. Let R and Θ be the radius and angle of the point. (a) Find the joint CDF of R and Θ. (b) Find the marginal CDFs of R and Θ. (c) Find the joint PDF of R and Θ. (d) Find the marginal PDFs of R and Θ. (e) Compute the probability that the point is in the first quadrant of the real plane and its radius exceeds 0.5