Expectation - May 20

[Amit Goyal]

May20_Expectation.png

[aditi5000]

My answer to part 2: (completely a shot in the dark)
The assignment which makes all propositions true:
x1, x2, x3, ~x4, x5, x6, ~x7, ~x8 and x9 need to be true.
In other words, x4, x7 and x8 need to be false.

[AJ]

A very basic doubt....

if x1 is false .. this implies ~x1 is true ....

right...????

[aditi5000]

Yep

[AJ]

Ummmm, I got,

(a)Expected no. of true propositions = 6.125

(b) Assignment such that all propositions are true... can be.. "when all x's are false"..

!!!! This sounds strange.. My answers are definitely wrong.... :(

[aditi5000]

Ok I solved this again. Now getting the minimum conditions required for all propositions to be true
x3,x6,x9 need to be true and,
either x4 needs to be false or  x7 needs to be false or x5 needs to be true (any one of them is enough)

As for expected values..
there are 7 propositions.
each has 0.5 chance of being true or false.
this depends on the truth value of the individual variables.
each of the 9 variables has .5 chance of being true and there are 3 variables in each proposition thus, .125 probability.
So expected no of true propositions
=7*0.5*9*0.125
=3.9375

[Amit Goyal]

Let me first ask you couple of basic questions.
Let S be the sample space.
Let P:Set of all events --> [0,1] be the probability function.
Let X: S --> R and Y: S --> R be two random variables.
Is Expectation(X + Y) = Expectation(X) + Expectation(Y) always true? If not, then give conditions under which it is true?

[Gayatri]

Yes.its always true that Expectation(X+Y)=Expectation(X)+Expectation(Y).

[anon_econ]

I don't know what's going on. i could only conclude that the probability of having zero true propositions is zero..lol. The only methods of proceeding that I can think of are almost as bad as listing down the 512 outcomes and classifying them. Amit sir please give a hint.

[AJ]

I think this is always true!

[Amit Goyal]

Well done. You are right. Its always true. Proof is elementary and you can do it on your own or refer any textbook for the same.
Let A ⊂ S be an event. Define
I[A]:S-->R
in the following way:
I[A](x) = 1 if x ∈ A
          = 0 if x ∉ A
What is Expectation(I[A])?

[aditi5000]

summation or integration x*f(x) depending on whether its discrete or continuous
So... E(A)=1 ?

[Amit Goyal]

I[A] is a random variable. And the question is: What is Expectation(I[A]), in short E(I[A])?

[anon_econ]

P(A)?
I'm probably interpreting it wrongly

[aditi5000]

Is it I[A] itself ?

[Amit Goyal]

Thats right. E(I[A]) = P(A)
Now lets come back to the problem. We have seven propositions based on 9 terms.
The first question is how many elements are there in the sample space? Give example of one element of a sample space.

[AJ]

Ya, It should be P(A) ...

from .. 1*P(A)+0*P(A')

I still don't get the question... :/

[anon_econ]

Umm. I'm not at all sure about this. The number of elements in the sample space is 2^7 i.e 128. An example is all propositions are true. Or maybe there r 2^9 i.e 512 elements, and an example is x1 is true, xi is false for all i not =1.

[anon_econ]

isn't it true that one experiment can have more than 1 sample space and u deal with the one that is relevant?

[anon_econ]

actually it can't be 2^7 bcoz if proposition 1 is not true then propositions 2 and 7 must be true