Reply to queries

[Aritra Mandal]

Hello learners,

I am Aritra Mandal, your TA. I have tried to address all the queries. 

Please go through the following. Feel free to reply if there is any doubt.

Question 1> I have a question about what is happening in this video at 00:11:18 -

http://www.youtube.com/watch?v=t6HO-QG-HhE#t=678s.

In the lecture video, sir has used S = {max{0,m-(N-r)}, ....... , min{m,r}}. The elements of S is the number of successes which is minimum 0 and atmost m. So, S' = {0,1,2,3, .... , m}. The definition of S used by sir gives the same set, S', if we evaluate. But I don't understand what is the motive behind using S not directly S', because S is somewhat complex than S', when we see. What is the use of S instead of S'? Is there any other use?.

Answer> Basically you deduce S' from S through computation. S' is the primitive form of the sample space. S' is described by all variables related to the Hypergeometric distribution distribution. Later in next step we simplify and get S.

Question 2> I am very confused in the multinomial distributiin and have the following doubts-

i) In the lecture, sir has defined Xj = No. of trials that result in the jth outcome. We have to find P(X1=x1, X2=x2,. ......, Xk=xk) where x1+x2+ .... +xk = n.  

So, here X1, ... , Xk are all random variables ( Xi is function from S to Ti) because sir has used notation X1=x1 meaning X1^{-1}(x1). I have a doubt why we have to define k random variables. We can define one random variable X such that each outcome in S is mapped to number of trials that result in jth outcome. Then our question of joint distribution which we have to find in the example looks like P(X=T) where T = range(X) = {x1,...,xk} instead of P(X1=x1, X2=x2,. ......, Xk=xk) ? I am not able to understand like how the k random variables Xi actually meant?   

ii) Also I have not clearly understood the set B(x1,...,xk).  

Answer> i) As you have said, we can can define one random variable X such that each outcome in S is mapped to number of trials that result in jth outcome. As a result for 'u' in sample space we have X(u)=(X1(u),X2(u),...,Xk(u)) .

Then P(X1=x1,X2=x2,...,Xk=xk)

= P({u in sample space| X1(u)=x1,X2(u)=x2,...,Xk(u)=xk})

= P({u in sample space|(X1,X2,...,Xk)(u)=(x1,x2,...,xk)})

= P({u in sample space| X(u)=(x1,x2,...,xk)})

= P(X=(x1,...,xk))

ii) B(x1,x2,..,xk) is an event defined by,

 B(x1,x2,..,xk)={u in sample space| X1(u)=x1,..,Xk(u)=xk}

Thanks and regards