Mat3 Probeklausur WiSe 22 23

1

a)

$$\Omega =\{1,...,6{\}}^3$$ $$\mathcal{A}=\{(x_1,x_2,x_3):x_1+x_2+x_3=11\vee x_1+x_2+x_3=11\}{x}_i\in \{1,...,6\}$$ $$\mathbb{P}(\mathcal{A})=\frac{|\mathcal{A}|}{|\Omega |}$$

b )

Permutationen von:

$$\begin{array}{rl}(A,B,C)& =|\{(A,B,C),(A,C,B),(B,A,C),(B,C,A),(C,A,B),(C,B,A)\}|=6\\ (A,A,B)& =|\{(A,A,B),(A,B,A),(B,A,A)\}|=3\\ (A,A,A)& =1\end{array}$$ $$\begin{array}{rl}A& =6+4+1\\ & =6+3+2\\ & =5+5+1\\ & =5+4+2\\ & =5+3+3\\ & =4+4+3\\ |A|& =6+6+3+6+3+3=27\end{array}$$ $$\begin{array}{rl}B& =6+5+1\\ & =6+4+2\\ & =6+3+3\\ & =5+5+2\\ & =5+4+3\\ & =4+4+4\\ |B|& =6+6+3+3+6+1=25\end{array}$$ $$|\Omega |=6^3=216$$ $$P(A)=\frac{|A|}{|\Omega |}=\frac{27}{216}P(B)=\frac{|B|}{|\Omega |}=\frac{25}{216}$$

c)

Nein da die Reinfolge relevant ist.

2

a)

$$\begin{array}{rl}{\int }_{-\infty }^{\infty }{f}_X(x)dx& =1\\ \gamma \cdot {\int }_1^{\infty }{x}^{-r}dx& =1\\ \gamma \cdot {\left[\frac{1}{1-r}{x}^{1-r}\right]}_1^{\infty }& =1\\ \gamma \left(0-\frac{1}{1-r}{1}^{1-r}\right)& =1\\ -\frac{\gamma }{1-r}& =1\\ \frac{\gamma }{r-1}& =1\\ \Rightarrow \gamma & =r-1\end{array}$$

b)

$$\begin{array}{rl}\mathbb{E}(x)& ={\int }_{-\infty }^{\infty }x\cdot f_X(x)dx\\ & =\gamma {\int }_1^{\infty }x\cdot x^{-3}dx\\ & =\gamma {\int }_1^{\infty }{x}^{-2}dx\\ & =\gamma {\left[\frac{1}{-1}{x}^{-1}\right]}_1^{\infty }\\ & =\gamma \left(0-\frac{1}{-1}\right)\\ & =\frac{\gamma }{1}\\ & =\gamma \end{array}$$

3

a)

$$\begin{array}{rl}y& =(y_1,...,y_n{)}^T\\ Z(y,\theta )& =\prod _{i=1}^n{f}_{\theta }(y_i)\\ & =\prod _{i=1}^n\frac{2y_i}{\theta }\exp \left(-\frac{y_i^2}{\theta }\right)\\ & ={\left(\frac{2}{\theta }\right)}^n\prod _{i=1}^n{y}_i\exp \left(-\frac{1}{\theta }\prod _{i=1}^n{y}_i^2\right)\\ L(y,\theta )& =\sum _{i=1}^{n}log\left(\frac{2y_i}{\theta }exp\left(-\frac{y_i^2}{\theta }\right)\right)\\ & =log\left({\left(\frac{2}{\theta }\right)}^n\right)+\sum _{i=1}^n{y}_i+log\left(exp\left(\frac{-1}{\theta }+\sum _{i=1}^n{y}_i^2\right)\right)\\ & =n\cdot log(2)-n\cdot log(\theta )+\sum _{i=1}^{n}log(y_i)+\frac{-1}{\theta }\sum _{i=1}^n{y}_i^2\\ \frac{dL(y,\gamma )}{d\theta }& =-\frac{n}{\theta }+\frac{1}{{\theta }^2}\sum _{i=1}^n{y}_i^2=0\\ \Rightarrow \frac{n}{\theta }& =\frac{1}{{\theta }^2}\sum _{i=1}^n{y}_i^2\\ n& =\frac{1}{\theta }\sum _{i=1}^n{y}_i^2\\ n\theta & =\sum _{i=1}^n{y}_i^2\\ \theta & =\frac{1}{n}\sum _{i=1}^n{y}_i^2\\ \Rightarrow {\hat{\theta }}_{ML}(y)& =\frac{1}{n}\sum _{i=1}^n{y}_i^2\end{array}$$