Statistica I
Esercizi preliminari: sommatorie e produttorie
Esercizio A
Si verifichi se le seguenti scritture sono in generale corrette:
\sum_{i=1}^na_i^2 = \left(\sum_{i=1}^n a_i\right)^2.
\sum_{i=1}^n\sum_{j=1}^ka_{ij}=\sum_{j=1}^k\sum_{i=1}^na_{ij}=\sum_{j=1}^n\sum_{i=1}^ka_{ji}.
\sum_{i=1}^n \sum_{j=1}^{k_i}a_{ij}=\sum_{j=1}^{k_i}\sum_{i=1}^na_{ij}.
\sum_{i=1}^n \sum_{j=1}^{k}a_{i}=n\sum_{i=1}^{n}a_{i}.
\sum_{i=1}^n\sum_{j=1}^{k}a_{i}=k\sum_{i=1}^{n}a_{i}.
\sum_{i=1}^n a_i=\sum_{i=m+1}^n a_i+\sum_{i=1}^m a_i.
\sum_{i=1}^n\alpha=\alpha.
\sum_{i=1}^n\alpha=0.
\sum_{i=1}^n\alpha = \sum_{i=5}^{n+4}\alpha.
\sum_{i=1}^n(a_i+b_i)^2=\sum_{i=1}^na_i^2+2\sum_{i=1}^n a_ib_i+\sum_{i=1}^nb_i^2.
\sum_{i=1}^n\frac{a_i}{2}=\frac{1}{2}\sum_{i=1}^na_i.
\sum_{i=1}^n\frac{2}{a_i}=2\frac{1}{\sum_{i=1}^n a_i}.
\sum_{i=1}^4(4+\pi+a_i)=16+4\pi +\sum_{i=1}^4a_i.
\sum_{i=1}^3(3a_i+2b_i)=3\sum_{i=1}^3a_i+2\sum_{i=1}^3 b_i.
\frac{\sum_{i=1}^na_ib_i}{\sum_{i=1}^nb_i}=\sum_{i=1}^na_i.
\log \left(\sqrt[n]{\prod_{i=1}^n a_i}\right)=\frac{1}{n}\sum_{i=1}^n \log a_i.
\sqrt[n]{\prod_{i=1}^n\left(\frac{a_i}{b_i}\right)}=\frac{\sqrt[n]{\prod_{i=1}^na_i}}{\sqrt[n]{\prod_{i=1}^nb_i}}.
Esercizio B
Determinare il valore delle seguenti somme
- \sum_{i=1}^4i^2.
- \sum_{i=3}^6i^2.
- \sum_{i=4}^7(i-1).
- \sum_{i=1}^5 6.
- \sum_{i=2}^5 6.
- \sum_{i=3}^6(i-1).
- \sum_{i=2}^4(5+3i+i^2).
- \sum_{i=1}^3\left(\frac{i^2}{4} + 3 i\right).
Esercizio C (opzionale, difficile)
Dimostrare che
\sum_{i=1}^n i = \frac{n(n+1)}{2}.