please show your step! thanks!

3. Suppose we observe N i.i.d data points D = {11, 12,..., IN}, where each In {1, 2, ..., K} is a random variable with categorical (discrete) distribution parameterized by 0 = (0,02, ...,Ok), i.e., Cat(61, 62, ..., Ok), n=1,2,..., N (8) In detail, this distribution means that for a specific n, the random variable In follows Pin = k) = 0k, k = 1, 2, ..., K. Equivalently, we can also write the density function of a categorical distribution as K P(In) = 11 e_e=k] (9) k=1 where I[In = k) is called identity function, and defined as 1, if In = k (10) 0, otherwise a. Now we want to prove that the joint distribution of multiple i.i.d categorical variables is a multinomial distribution. Show that the density function of D = {11, 12, ..., UN} is I[an= k] = { K p(D\0) = ITO (11) k=1 where Nx = N-LI[Tn = k) is the number of random variables belonging to category k. In other word, D = {11, 12, ..., In} follows a multinomial distribution. 3. Suppose we observe N i.i.d data points D = {11, 12,..., IN}, where each In {1, 2, ..., K} is a random variable with categorical (discrete) distribution parameterized by 0 = (0,02, ...,Ok), i.e., Cat(61, 62, ..., Ok), n=1,2,..., N (8) In detail, this distribution means that for a specific n, the random variable In follows Pin = k) = 0k, k = 1, 2, ..., K. Equivalently, we can also write the density function of a categorical distribution as K P(In) = 11 e_e=k] (9) k=1 where I[In = k) is called identity function, and defined as 1, if In = k (10) 0, otherwise a. Now we want to prove that the joint distribution of multiple i.i.d categorical variables is a multinomial distribution. Show that the density function of D = {11, 12, ..., UN} is I[an= k] = { K p(D\0) = ITO (11) k=1 where Nx = N-LI[Tn = k) is the number of random variables belonging to category k. In other word, D = {11, 12, ..., In} follows a multinomial distribution