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Likelihood equations and Complex Analysis help for rep watch

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    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 1+z+z^2+ .... + z^n= \dfrac{1-z^{n+1}}{1-z} with z as a complex number. Derive

    1+\cos(\theta)+\cos(2\theta)+...  .+\cos(n\theta) = \dfrac{1}{2}+\frac{\sin((n+\frac  {1}{2})\theta)}{2\sin(\frac{\the  ta}{2})}
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    3. Set z = \cos \theta + i \sin \theta in your given equation and take the real part of the right-hand-side.
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    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
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    If z=e^{i\theta}, the simplest thing is to multiply the top and bottom of the RHS fraction by e^{-i\theta/2}, since this leaves e^{-i\theta/2} - e^{i\theta/2} as the denominator, which is real. (And so you don't have to play the usual games with multiply by the conjugate etc).
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    That sounds good to me. Thanks!
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    Just bumping this up. If anyone's got more help on 1 and 2 that would be great
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    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.
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    Thanks and I am at university yeah I'll follow up those hints and see how I get on.
 
 
 
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