Expectation - May 20

[AJ]

But we can't use binomial .. cuz these propositions are not independent.. although x's are!!
Haina?

[anon_econ]

I'll think about it later. My mind isn't working now. Where has sir gone? He must have got tired of spoon feeding us. We should have got it as soon as he wrote N=I(E1)+..

[anon_econ]

u r the one who used to post on alex's blog too, right?

[AJ]

I knowwwww.....

Ya, do think about it.. I mean if this is the answer... How come the binomial thing also gave the same answer..???

[Amit Goyal]

Cool. So, E(N) = 7(7/8) = 6.125. And you all know how. Well done.
Now do part (ii). Remember that you have to do it using (i).

[anon_econ]

Thanks sir :)
I learnt a few new things today :)
I will try part 2 later..my mind is seriously shut. AJ tum lage raho..
n yea AJ v interacted there :)

[AJ]

@sir..
Why I got the same answer using binomial theorem.. which was wrong method..?
pure coincidence ?

[Amit Goyal]

AJ that happened by chance that the binomial thing also gave you the same answer. But your method was incorrect.
Let me tell you why?
Consider 21 terms instead of 9.
x(1), x(2), x(3), ...., x(21).
Let the seven propositions be:
P(1):  x(1) v x(2) v x(3)
P(2):  x(4) v x(5) v x(6)
...
P(j):   x(3j - 2) v x(3j - 1) v x(3j)
...
P(7): x(19) v x(20) v x(21)
Each of the 21 takes values T and F with probability 0.5 and independent. Note that here all truthfulness of all propositions are also independent.

Now the procedure that we did just now and what you did will give you the same answer. In this case its binomial distribution because each proposition takes values true or false independently. So you can use your method here. And you were lucky to get the right number in the earlier problem.

[AJ]

okay! I got it now... so binomial method was incorrect because propositions were not independent!

and for (b) part....
we have E(N)=6.125

Say, we have P(N=7)=0 .. (i.e. there is no such assignment that makes all propositions true)

then E(N) can never come out to be more than 6.. at max it will be 6...

So, if we are having expectation of N above 6 .. this implies (N=7) has some probability...
i.e. Atleast one such assignment exists!

I don't know how to write this mathematically!

And, this sure was a good day sir! :)

Thankyou sooo much!

[Amit Goyal]

Well done AJ :)

[AJ]

Thankyou! :)

[SHIKHA]

AJ
i didnt get how p=7/8...???