1.Let X,Y,Z follow a trinomial distribution with success probabilities px,py,pz >0such that px+py+pz=1 and sample size n∈N.(a)Write down the expected valuesμx=E[X]andy=E[Y].[2](b)Find the covariances Cov(X,X),Cov(X,Y)and Cov(Y,Y).[3]YP:STAT0005,2021-202214(c)Let V,W follow a bivariate normal distribution such that E[V=EX],EW]=E[Y,Var(V)Var(X),Var(W)=Var(Y)and Cov(V,W)=Cov(X,Y).Fork∈{O,l,2,.,n},compute the expected value E[XY=k]as well as theexpected value EVW=k and compare them.[4](d)Fork∈{O,l,2,.,n},compute the conditional variances a=Var(X|Y=k)and b Var(VW k).Consider whether or not a and b can be equal,asfollows:i.If you think that equality of a and b is independent of the values k,py,n,write down the conditions under which the normal distribution can be usedas an approximation to the binomial distribution and discuss whether theyare satisfied for the conditional distribution of V given W=k.ii.If you think that equality of a and b depends on the values of k,py,n,identifythe values of k,py,n for which the variances agree and comment on them.

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