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# help !!! generating function

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1. If N is number of heads in n fair coin tosses, then show that
the generating function G X(s) = (2^(-n) )(1+s)^n
2. (Original post by vanessa_tram)
If N is number of heads in n fair coin tosses, then show that
the generating function G X(s) = (2^(-n) )(1+s)^n
What have you tried?
3. for my other exercises i've been doing, I think it should be (see attached file)
but then i dont know what to do next. i dont know if i should have E(s^N/tails)P(tails) in there or not as we only want number of heads.
Attached Images

4. (Original post by Zacken)
What have you tried?
for my other exercises i've been doing, I think it should be (see attached file)but then i dont know what to do next. i dont know if i should have E(s^N/tails)P(tails) in there or not as we only want number of heads.
5. (Original post by vanessa_tram)
for my other exercises i've been doing, I think it should be (see attached file)but then i dont know what to do next. i dont know if i should have E(s^N/tails)P(tails) in there or not as we only want number of heads.
No need to condition on anything here. The probability distribution for the number of heads thrown is Binomial with parameters n and 1/2. You then just calculate the sum resulting from the definition of a generating function.
6. (Original post by Gregorius)
No need to condition on anything here. The probability distribution for the number of heads thrown is Binomial with parameters n and 1/2. You then just calculate the sum resulting from the definition of a generating function.
thank you, i got to here, do u know how to sum this up ???
7. (Original post by vanessa_tram)
thank you, i got to here, do u know how to sum this up ???
Do you know how to sum ?

Think about the binomial series with a certain value of .
8. (Original post by Zacken)
Do you know how to sum ?

Think about the binomial series with a certain value of .
i used this theorem so i got (1+(1/2)s)^n , still cant get the answer
9. (Original post by vanessa_tram)
i used this theorem so i got (1+(1/2)s)^n , still cant get the answer
Huh? for some suitable .
10. (Original post by Zacken)
Huh? for some suitable .
i got it wrong from the last step cos it was power of n, not x, so i cant use that formula, i dont get ur hint, sorry, please help me again, please. can i write like this ?
11. (Original post by vanessa_tram)
i got it wrong from the last step cos it was power of n, not x, so i cant use that formula, i dont get ur hint, sorry, please help me again, please. can i write like this ?
is just a constant, so you can pull it out of your sum.

Then you're left with

where the sum evaluates to , then multiply it by the constant.
12. (Original post by vanessa_tram)
thank you, i got to here, do u know how to sum this up ???
Your equation isn't quite right yet. Remember that the definition of the generating function is

So you want

Not an as you have in your formula.
13. (Original post by Zacken)
is just a constant, so you can pull it out of your sum.

Then you're left with

where the sum evaluates to , then multiply it by the constant.
but as we have s^n outside, no matter how we expand inside the bracket, how can we get the result with (1+s)^n
14. (Original post by vanessa_tram)
but as we have s^n outside, no matter how we expand inside the bracket, how can we get the result with (1+s)^n
15. (Original post by Zacken)
thank you, i got the answer, made a little mistake , he just pointed it out
16. (Original post by Gregorius)
Your equation isn't quite right yet. Remember that the definition of the generating function is

So you want

Not an as you have in your formula.
thank you very much, i got it now.
17. (Original post by vanessa_tram)
thank you, i got the answer, made a little mistake , he just pointed it out
Don't thank me, I was essentially useless - Greg can do with some praise
18. (Original post by Zacken)
Don't thank me, I was essentially useless - Greg can do with some praise
no, thank you u was trying ur best to help me.
19. (Original post by Zacken)
Don't thank me, I was essentially useless - Greg can do with some praise
That's OK, just throw rose petals and money...
20. Oh God I bet this is A-level Maths/FM isn't it... I'm going to cry...

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