We're interested in the relationship between two random variables X{0,1} and Y{0,1}. Specifically, we're interested in something called the "odds ratio." Define following notation: p(y,x)=P(Y=y,X=x)p(yx)=P(Y=yX=x) Suppose that p(y,x)>0 for all possible combinations of (y,x). Then, we'll define the odds ratio as: OR=p(00)p(01)p(10)p(11) (a) Express OR in terms of p(0,0),p(0,1),p(1,0), and p(1,1). (b) Suppose we have a sample (Y1,X1),,(Yn,Xn) that are iid (X,Y). Define p^n(y,x)ORn^=n1i=1n1{Yi=y,Xi=x}=p^n(0,0)p^n(0,1)p^n(1,0)p^n(1,1) Show that ORn^ is a consistent estimator for OR. (c) What will np^n(1,1)p^n(1,0)p^n(0,1)p^n(0,0)pn(1,1)pn(1,0)pn(0,1)pn(0,0) converge to in distribution as n ? (The things in square brackets are 41 column vectors). Make sure the variance of the limiting distribution is specified (a general element-wise description is sufficient - not need to lay out the entire matrix). (Hint: look at the multivariate version of one of our familiar statistics results). We're interested in the relationship between two random variables X{0,1} and Y{0,1}. Specifically, we're interested in something called the "odds ratio." Define following notation: p(y,x)=P(Y=y,X=x)p(yx)=P(Y=yX=x) Suppose that p(y,x)>0 for all possible combinations of (y,x). Then, we'll define the odds ratio as: OR=p(00)p(01)p(10)p(11) (a) Express OR in terms of p(0,0),p(0,1),p(1,0), and p(1,1). (b) Suppose we have a sample (Y1,X1),,(Yn,Xn) that are iid (X,Y). Define p^n(y,x)ORn^=n1i=1n1{Yi=y,Xi=x}=p^n(0,0)p^n(0,1)p^n(1,0)p^n(1,1) Show that ORn^ is a consistent estimator for OR. (c) What will np^n(1,1)p^n(1,0)p^n(0,1)p^n(0,0)pn(1,1)pn(1,0)pn(0,1)pn(0,0) converge to in distribution as n ? (The things in square brackets are 41 column vectors). Make sure the variance of the limiting distribution is specified (a general element-wise description is sufficient - not need to lay out the entire matrix). (Hint: look at the multivariate version of one of our familiar statistics results)