pdf of area and circumference of a circle

May 2010
11
0
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
Dec 2007
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Zeitgeist
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?
 
May 2010
11
0
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
Dec 2007
16,948
6,768
Zeitgeist
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.
 
May 2010
11
0
how did you find ? is it the same for c?
 

matheagle

MHF Hall of Honor
Feb 2009
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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 +?