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Glen_b
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Mean and Variance of the Area of a Circle w/with Uniform Radius

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whuber ♦
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A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.
(a) Find the mean and variance of A, without first finding the CDF or PDF of A.
(b) Find the CDF and PDF of A.

So, quite obviously, we know the equation of a circle as A = ${\pi}r^2$. If we are given a uniform distribution, can we not just find E(R)$E(R)$ and then, using that value, find E(A)$E(A)$, and likewise for the variance? And, if we have F(A)$F(A)$, is this the CDF or PDF?

PLEASE help. Thanks!

A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.
(a) Find the mean and variance of A, without first finding the CDF or PDF of A.
(b) Find the CDF and PDF of A.

So, quite obviously, we know the equation of a circle as A = ${\pi}r^2$. If we are given a uniform distribution, can we not just find E(R) and then, using that value, find E(A), and likewise for the variance? And, if we have F(A), is this the CDF or PDF?

PLEASE help. Thanks!

A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.
(a) Find the mean and variance of A, without first finding the CDF or PDF of A.
(b) Find the CDF and PDF of A.

So, quite obviously, we know the equation of a circle as A = ${\pi}r^2$. If we are given a uniform distribution, can we not just find $E(R)$ and then, using that value, find $E(A)$, and likewise for the variance? And, if we have $F(A)$, is this the CDF or PDF?

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Sophie B.
Sophie B.
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Mean and Variance of Circle w/ Uniform Radius

A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.
(a) Find the mean and variance of A, without first finding the CDF or PDF of A.
(b) Find the CDF and PDF of A.

So, quite obviously, we know the equation of a circle as A = ${\pi}r^2$. If we are given a uniform distribution, can we not just find E(R) and then, using that value, find E(A), and likewise for the variance? And, if we have F(A), is this the CDF or PDF?

PLEASE help. Thanks!