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Consider this question and the working below:

A coin-making machine produces pennies. Each penny is manufactured to have a probability $P$ of turning up heads. However, the machine draws P randomly from the uniform distribution on [0,1] so $P$ can differ for each coin produced. A coin pops out of the machine. You flip it once, and it comes up heads. Given this information, what is the conditional distribution function or the probability of a head for that coin?

We have

$$P \sim U[0,1]$$

The probability distribution function:

$$f_P(p) = 1 \qquad (0\le p \le 1)$$

Let $X$ denote the random variable that the first toss is a head ($X=1$ in this case).

$$X|(P=p) \sim \text{Bernoulli}(p)$$

The probabiity mass function:

$$P(X=x|P=p) = p^x(1-p)^{1-x} \qquad (0\le p \le 1 \text{ and } x=0,1)$$

We need $f_{P|X}(p|x)$.

Get the joint distribution first:

$$P(X=x|P=p) = \frac{f_{X,P}(x,p)}{f_P(p)} \implies f_{X,P}(x,p) = P(X=x|P=p)f_P(p)$$ $$ f_{X,P}(x,p) = p^x(1-p)^{1-x} \qquad (0\le p \le 1 \text{ and } x=0,1)$$

I am not sure if this is the correct joint distribution? Is the relation I used to get it valid?

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1 Answer 1

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Using Bays $$P(P|H)P(H)=P(H|P)P(P)$$

Solving for P(P|H) $$P(P|H)=\frac{P(H|P)}{P(H)}*P(P)$$

Converting to distributions $$f_{P,X}(p|x=1)=\frac{P(H|P)}{P(H)}*f_P(p)$$

P(H|P) is just p, P(H) = 1/2, f_P(p) = 1 $$f_{P,X}(p|x=1)=\frac{p}{\frac{1}{2}}*f_P(p)$$

$$f_{X,P}(p|x=1)=2p$$

Checking, to find the Expected Value, E(P|H) $$E(P|H)=\int_0^1{p*f_P(p|H) dp}$$

$$E(P|H)=\int_0^1{p*2p dp}$$

$$E(P|H)=\int_0^1{2p^2 dp}$$

$$E(P|H)=\frac{2}{3}p^3|_{p=0}^{p=1} $$

$$E(P|H)=\frac{2}{3}$$

This makes sense to me, at least :)

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  • $\begingroup$ I think this is correct, but I want to understand where my approach is going wrong. Can you see why I am missing a factor of 2 in my question? $\endgroup$ Apr 15, 2016 at 21:26
  • $\begingroup$ Thanks! The factor of 2 comes from the a priori estimate of the probability of heads, a P(X=x) term. I think it should be on the lps on the second equation line after "get the joint distribution first." $\endgroup$
    – MikeP
    Apr 16, 2016 at 10:41
  • $\begingroup$ So are saying that $P(X=x|P=p)$ (in my notation) should have a 2? $\endgroup$ Apr 16, 2016 at 13:03
  • $\begingroup$ Actually, I may have messed up. I need to think on this a bit $\endgroup$
    – MikeP
    Apr 17, 2016 at 0:21
  • $\begingroup$ Sorry for the delay! Actually it's the $f_{X,P}(x,p)$ that needs the factor of 2, otherwise it won't integrate to 1 which is necessary for a valid pdf $\endgroup$
    – MikeP
    Apr 17, 2016 at 14:54

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