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probility X is a head be obtained.and the distribution of X in a large collection of coins is specified by this p.d.f: f(x)=6x(1-x) for x(0,1), 0 others.a coin is selected at random and tossed once,it is head.the conditional p.d.f. of X for this coin?I wonder if I use the f(x,p)=g(p/x)*f(x) for x in(0,1) and p in[0,1], which is the binomail distribution?f(x,p) or g(p/x)?

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    we can calculate the f(x,p) first, and get the g(x/p) after.I guess.2012-06-24

2 Answers 2

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Let $A$ be the event that the coin turns out heads. We know that:

$P(A|X=x) = x$

And wish to determine $f(x|A)$. You should use Bayes' Theorem, that is,

$f(x|A) \propto f(x)P(A|X=x)$

Plugging in the values:

$f(x|A) \propto 6x^2(1-x)$

From the non-constant part $x^{2}(1-x)$ you can conclude that $f(x|A)$ is a beta with parameter $(3,2)$.

Another way to do this is to recall that a coin flip is given by a Bernoulli random variable variable and that $f(x)$ is the distribution of a Beta$(2,2)$. Since the beta is a conjugate prior for the Bernoulli, conclude that $f(x|A)$ is the density of a Beta $(3,2)$.

A third way is to really use the exact expression for Bayes's rule:

$f(x|A) = \frac{f(x)P(A|x)}{\int_{0}^{1}{f(x)P(A|x)}}$

$f(x|A) = \frac{6x(1-x)x}{\int_{0}^{1}{6x(1-x)x}}$

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I think I must figure the joint p.d.f first, and use this to calculate the conditional p.d.f of X.

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