Question

A point is uniformly distributed within the disk of radius 1 . That is, its density is $$ f(x, y)=C, \quad 0 \leq x^{2}+y^{2} \leq 1 $$ Find the probability that its distance from the origin is less than \(x, 0 \leq x \leq 1\).

Short answer

The probability that a randomly chosen point in the disk of radius 1 lies within a distance less than \(x\) from the origin is given by \(P(r < x) = x^{2}\) for \(0 \leq x \leq 1\).

Step-by-step solution

State the given density function

We are given the density function f(x, y) as: \( f(x, y) = C \), for \(0 \leq x^{2} + y^{2} \leq 1\) #Step 2: Find the value of the constant C#

Integrate the density function over the entire disk

We need to find the value of C that makes the density function valid. To do this, we integrate the density function over the entire disk of radius 1 and set the result equal to 1 (since the total probability should be 1). Thus, we have: \( \int_{0}^{2\pi}\int_{0}^{1}C r \, dr \, d\theta = 1 \) #Step 3: Calculate the value of C#

Calculate the integrals and solve for C

We evaluate the integral: \( C \int_{0}^{2\pi}d\theta \int_{0}^{1}r \, dr = 1 \) \( C \cdot 2\pi \cdot \frac{1}{2} = 1 \) Now, solve for C: \( C = \frac{1}{\pi} \) Thus, our density function is given by: \( f(x, y) = \frac{1}{\pi} \) for \( 0 \leq x^{2} + y^{2} \leq 1\) #Step 4: Compute the probability of distance less than x from the origin#

Integrate the density function over the desired region

We need to find the probability that a point lies within a distance less than x from the origin. To do this, we integrate the density function over the region defined by \(0 \leq x^{2} + y^{2} \leq x^{2}\): \( P(r < x) = \int_{0}^{2\pi}\int_{0}^{x}\frac{1}{\pi} r \, dr \, d\theta \) #Step 5: Evaluate the integral#

Calculate the integrals

We evaluate the integral: \( P(r < x) = \frac{1}{\pi} \int_{0}^{2\pi}d\theta \int_{0}^{x}r \, dr \) \( P(r < x) = \frac{1}{\pi} \cdot 2\pi \cdot \frac{x^{2}}{2} \) #Step 6: Simplify the expression for the probability#

Simplify the result

After simplifying the expression, we get the probability that a point lies within a distance less than x from the origin: \( P(r < x) = x^{2} \) for \(0 \leq x \leq 1\) This is the final expression for the probability that a randomly chosen point in the disk of radius 1 lies within a distance less than x from the origin.