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![[tex]P(X=x)=\frac{e^{-u}u^x}{x!}[/tex]](images/latex/b1c7b3c41223112a96bb51d9e1526d5a9f3b0008_0.png "[tex]P(X=x)=\frac{e^{-u}u^x}{x!}[/tex]")
![[tex]u[/tex]](images/latex/c0a8c2cb7cac5d96b201825d488d0e85101ddfa1_0.png "[tex]u[/tex]")
para el cual el caso de la muestra tenga la mayor probabilidad de haber salido:
![[tex]A=\mathop{\prod}_{i = 1}^{n } P_{x}(x_i)=\mathop{\prod}_{i = 1}^{n }\frac{e^{-u}u^x_i}{x_i!}[/tex]](images/latex/9c5bbab23a4800d85cadf49bbc4aa5d1c5336cf8_0.png "[tex]A=\mathop{\prod}_{i = 1}^{n } P_{x}(x_i)=\mathop{\prod}_{i = 1}^{n }\frac{e^{-u}u^x_i}{x_i!}[/tex]")
![[tex]A[/tex]](images/latex/734ef645c78704dbe5a136258a49d8425298f623_0.png "[tex]A[/tex]")
e igual al maximo de ![[tex]ln(A)[/tex]](images/latex/7f086388ca64a045f9081d974faaa1fec673f4d8_0.png "[tex]ln(A)[/tex]")
![[tex]L=ln\left(A\right)=\mathop{\sum}_{i = 1}^{n } ln\left(\frac{e^{-u}u^{x_i}}{X_i!}\right)=\mathop{\sum}_{i = 1}^{n }[ln\left(e^{-u}\right)+x_i ln\left(u^{x_i}\right)-ln\left(x_i!\right)][/tex]](images/latex/bfa8e0ce0efc4643fb85e00b9f910c82b2e2c9de_0.png "[tex]L=ln\left(A\right)=\mathop{\sum}_{i = 1}^{n } ln\left(\frac{e^{-u}u^{x_i}}{X_i!}\right)=\mathop{\sum}_{i = 1}^{n }[ln\left(e^{-u}\right)+x_i ln\left(u^{x_i}\right)-ln\left(x_i!\right)][/tex]")
![[tex]=\mathop{\sum}_{i = 1}^{n }[(-u)+x_iln(u)-ln(x_i!)][/tex]](images/latex/bc8943bab538253776b982204f719f8c804e9c25_0.png "[tex]=\mathop{\sum}_{i = 1}^{n }[(-u)+x_iln(u)-ln(x_i!)][/tex]")
![[tex]=-nu+ln(u)\mathop{\sum}_{i=1}^{n}x_i-\mathop{\sum}_{i=1}^{n}ln(x_i!)[/tex]](images/latex/f8504998ce9361de1ba3b7d33ad04de6e4684228_0.png "[tex]=-nu+ln(u)\mathop{\sum}_{i=1}^{n}x_i-\mathop{\sum}_{i=1}^{n}ln(x_i!)[/tex]")
![[tex]\frac{dL}{du}=-n+\frac{\mathop{\sum}_{i=0}^{n}x_i}{u}=0[/tex]](images/latex/e26673ba6dd49bc168f5191b049ddaf1c4893763_0.png "[tex]\frac{dL}{du}=-n+\frac{\mathop{\sum}_{i=0}^{n}x_i}{u}=0[/tex]")
![[tex]\Rightarrow\hat u=\frac{\mathop{\sum}_{i=0}^{n}x_i}{n}=\overline X[/tex]](images/latex/9f6d2c5d6d5807c90347bda17f1f0e1e35f1e6e7_0.png "[tex]\Rightarrow\hat u=\frac{\mathop{\sum}_{i=0}^{n}x_i}{n}=\overline X[/tex]")
![[tex]u=T\lambda[/tex]](images/latex/0074470ad1286fbb570cfd5fbf6929a2e3410767_0.png "[tex]u=T\lambda[/tex]")
![[tex]\Rightarrow \hat\lambda=\frac{\hat u}{T}[/tex]](images/latex/ee9d46a400e37641e431c117cffc11a181893be6_0.png "[tex]\Rightarrow \hat\lambda=\frac{\hat u}{T}[/tex]")
![[tex]T=1[/tex]](images/latex/f5faa500f18076bd3e942e5085a5c1fa610e46ff_0.png "[tex]T=1[/tex]")
, ![[tex]\hat \lambda=\frac{7+6+5+8+12}{5}=7,6[/tex]](images/latex/ae5dbf5af64650cf2ede983826da94f0f45c0e94_0.png "[tex]\hat \lambda=\frac{7+6+5+8+12}{5}=7,6[/tex]")
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