о о о Ô MLE Ô n Σ xi i=1 n MLE = = II xi! i=1 none of the other answers is correct 1 MLE == n ገራ Σ Ꮖ ; Ô О ○ Ô MLE = - по i=1 A discrete random variable X with the range (i.e. the set of all possible values) Rx = {0, 1, 2, 3,... } has probability mass function f(x; 0) = -0.0 e- a! x = 0, 1, 2, 3,... where is an unknown parameter. We want to find maximum likelihood estimate MLE of the parameter based on a sample (x1, x2,...,xn)? To do that, we find the likelihood function n L(0) = L(0; x1, . . ., x n ) = II ƒ (xi ; 0) · = i=1 = i=1 n Oxi e 0x1 xi! e-o 0x2 e-o Өх x2! xn! xi! -по x₁! =en. ΑΣ 12 Then, the log-likelihood function is In L(0) = ln L(0; x1,...,xn) = lne¯no . n II i=1 In en+In+In ( 1 · II 1 xi ! n =-10+(Σ Σπι)· .In In i=1 1 ·II xi! To find Ô MLE (the maximum likelihood estimator of 0), we maximize In L(0) while thinking of it as a function of only, i.e. treating xi's as given constants, and thus, treating the last term In ( 1 I_! as a constant. Eventually, what do we get as the maximum likelihood estimator MLE of the parameter 0?

о о о Ô MLE Ô n Σ xi i=1 n MLE = = II xi! i=1 none of the other answers is correct 1 MLE == n ገራ Σ Ꮖ ; Ô О ○ Ô MLE = - по i=1 A discrete random variable X with the range (i.e. the set of all possible values) Rx = {0, 1, 2, 3,... } has probability mass function f(x; 0) = -0.0 e- a! x = 0, 1, 2, 3,... where is an unknown parameter. We want to find maximum likelihood estimate MLE of the parameter based on a sample (x1, x2,...,xn)? To do that, we find the likelihood function n L(0) = L(0; x1, . . ., x n ) = II ƒ (xi ; 0) · = i=1 = i=1 n Oxi e 0x1 xi! e-o 0x2 e-o Өх x2! xn! xi! -по x₁! =en. ΑΣ 12 Then, the log-likelihood function is In L(0) = ln L(0; x1,...,xn) = lne¯no . n II i=1 In en+In+In ( 1 · II 1 xi ! n =-10+(Σ Σπι)· .In In i=1 1 ·II xi! To find Ô MLE (the maximum likelihood estimator of 0), we maximize In L(0) while thinking of it as a function of only, i.e. treating xi's as given constants, and thus, treating the last term In ( 1 I_! as a constant. Eventually, what do we get as the maximum likelihood estimator MLE of the parameter 0?