pdf of area and circumference of a circle

uva123

Suppose that the radius X of a circle is a random variable having the following p.d.f.:
f(x)={ (1/8)(3x=1) for 0<x<2
0 otherwise
Determine the p.d.f. of the area of the circle and the circumference of the circle.

can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???

mr fantastic

MHF Hall of Fame
Suppose that the radius X of a circle is a random variable having the following p.d.f.:
f(x)={ (1/8)(3x=1) for 0<x<2
0 otherwise
Determine the p.d.f. of the area of the circle and the circumference of the circle.

can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???
You need to find the pdf of the random variables:

1. $$\displaystyle A = \pi X^2$$

2. $$\displaystyle C = 2 \pi X$$

where it's known that the pdf of X is $$\displaystyle f(x) = \frac{3x + 1}{8}$$ for 0 < x < 2 and 0 otherwise.

You do NOT make a direct substitution. One way of doing it is to first find the cdf of each and then differentiate. What have you been taught about transforming random variables?

uva123

i havent learned about cpf's yet. i i cant figure our how to find the pdf of the area and circumference. that is where i am getting stuck.
thanks for you help!

mr fantastic

MHF Hall of Fame
i havent learned about cpf's yet. i i cant figure our how to find the pdf of the area and circumference. that is where i am getting stuck.
thanks for you help!
The cdf of A is given by:

$$\displaystyle G(a) = \Pr(A < a) = \Pr(\pi X^2 < a) = \Pr\left( - \frac{\sqrt{a}}{\sqrt{\pi}} < X < \frac{\sqrt{a}}{\sqrt{\pi}}\right) = \int_{0}^{\frac{\sqrt{a}}{\sqrt{\pi}}} f(x) \, dx = ....$$

for $$\displaystyle 0 < a < 4 \pi$$

and so the pdf of A is given by $$\displaystyle g(a) = \frac{dG}{da} = ....$$

The details are left for you. The pdf of C can be found in a similar way.

uva123

how did you find ? is it the same for c?

matheagle

MHF Hall of Honor
Suppose that the radius X of a circle is a random variable having the following p.d.f.:
f(x)={ (1/8)(3x=1) for 0<x<2
0 otherwise
Determine the p.d.f. of the area of the circle and the circumference of the circle.

can anyone help me with this one? do i just plug f(x) into the equations for area and circumference? how does that apply to the pdf of area and circumference???

is that = a +?

mr fantastic

MHF Hall of Fame
how did you find ? is it the same for c?
What restriction does the support of X place on the support of A? On C?