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# Likelihood equations and Complex Analysis help for rep watch

1. I've 3 questions which don't give me any joy currently...

1. A lecturer reckons the probability that any one of his 10 students will pass a test is theta, the same for any student. In the event, all ten students passed.

Suppsing these were independent tests, write down and sketch the likelihood function (there's more but i think i can continue from there with your help)

2. There are n fish in a pond. 3 are caught, tagged and thrown back in. Then a sample of 2 is captured which contains just one tagged fish. The parameter is n and is discrete.

Assume there is random mixing so that each sample is equally probably. How many ways are there of selecting 2 items from n? Evaluate the likelihood function of n.

3. Application of De Moivre's Theorem:

given with z as a complex number. Derive

2. 3. Set in your given equation and take the real part of the right-hand-side.
3. I've played around with the RHS for a bit but it's the trig identities I am stuck with. i don't know which one to get down to the answer Thanks though for replying
4. If , the simplest thing is to multiply the top and bottom of the RHS fraction by , since this leaves as the denominator, which is real. (And so you don't have to play the usual games with multiply by the conjugate etc).
5. That sounds good to me. Thanks!
6. Just bumping this up. If anyone's got more help on 1 and 2 that would be great
7. 1. The probability of one student passing is p, which is a bernoulli trial. 10 independent bernoulis is simply a binomial (10,p). It's only asking for a sketch, any idea what shape it would take? (I've used p instead of theta). Which value of p do you think would give the the highest likelihood?

2. This looks like a multinomial (which if you're doing A level is not studied??) but you're only choosing 2 fish with 1 tagged so it's not as complicated. Assuming they want the likelihood equation of choosing 2 fish from n and having 1 tagged, write the probabilities out for this event eg prob picking 1 tagged from n is just 3/n, followed by a non-tagged is (n-3)/(n-1) (there is one more case). With parameter n, this will be your likelihood function.

I am quite rusty with probability, but if you did it as a multinomial it should give you the same answer as above with: 3C1.(n-3)C1/nC2 (where nCr is n choose r, i cant remember latex).

I dont know what is meant by 'evaluate the likelihood function', perhaps they mean just to find it.
8. Thanks and I am at university yeah I'll follow up those hints and see how I get on.

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