Random Number Generator

[Kreizhn]

Homework Statement

Say you have two biased random number generators that will spit out the numbers 1, 2, 3. They're both biased: the first one has a distribution of $(\frac{1}{2}, \frac{1}{3}, \frac{1}{6} )$ and the second one has a distribution of $(\frac{1}{2}, \frac{1}{2}, 0 )$. Now let's say that somebody flips a fair coin (50/50 odds), if heads then they use the first number generator, if tails the second.

Write down an observation (as a set of indicator functions), such that the observation has only two possible outcomes, and that gives you the best chance of guessing which generator was used. What is the possibility of guessing correctly both before and after the observation.

Homework Equations

The probability distribution of a 1,2,or 3 appearing is simply
$$\rho = \displaystyle \frac{1}{2} \rho_1+ \frac{1}{2} \rho_2$$
$$= \displaystyle \frac{1}{2} \left( \frac{1}{2}, \frac{1}{3}, \frac{1}{6} \right) + \frac{1}{2} \left( \frac{1}{2} , \frac{1}{2}, 0 \right)$$
$$= \displaystyle \frac{1}{12} \begin{pmatrix} 6 \\ 5 \\ 1 \end{pmatrix}$$

The Attempt at a Solution

Again, really not sure how to proceed.

 

[lippyka]

Any help would be greatly appreciated. My guess is that the best observation would be the number of times a 1 appears in the sequence of numbers generated by the random number generator. If the first generator was used, then the probability of observing a 1 would be \frac{1}{2}. If the second generator was used, then the probability of observing a 1 would be \frac{1}{2}. Thus, the probability of guessing correctly before the observation is 0.5 and the probability of guessing correctly after the observation is also 0.5.