to [ useful modification of the -means algorithm is the fuzzy -means algorithmseee [ popular way to choose the starting positions in -means is given by the -means+heuristicintroduced in [ exercises this exercise is to show that the fisher information matrix (thin ( is equal to the matrix (thin ( )in the special case where (th)and under the assumption that integration and differentiation orders can be interchanged quotient rule for differentiation (alet be vector-valued function and real-valued function prove the following quotient rule for differentiation[ (th)/ (th) (th (th (th)th (thth (thth ( thand (thg( thin ( and take expectations with (bnow take (thg( th respect to eth on both sides to show that th ( th - (th- (theth ( thth { (cfinally show that is the zero matrix plot the mixture of ( ) ( )and exp( distributionswith weights / denote the pdfs in exercise by respectively suppose that is simulated via the two-step procedurefirstdraw from { }then draw from fz how likely is it that the outcome of has come from the uniform pdf simulate an iid training set of size from the gamma( distributionand implement the fisher scoring method in example to find the maximum likelihood estimate plot the true and approximate pdfs let { xn be iid data from pdf ( thwith fisher matrix (thexplain whyunder the conditions where ( holdsn st (th: (xi thn = for large has approximately multivariate normal distribution with expectation vector and covariance matrix (th)/ figure shows gaussian kde with bandwidth on the points - and reproduce the plot in python using the same bandwidthplot also the kde for the same databut now with ph( / [-