Analisi di DUE dadi

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Osserva

  1. Ciascuna faccia di dado ha probabilità
    p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = \displaystyle \frac{1}{6}
  2. Lanciando due dadi si ottengono i risultati della tabella a destra, dove sono evidenziate le combinazioni per il 7
    p(7) = \displaystyle \frac{6}{36} = \displaystyle \frac{1}{6}
  3. Dalla tabella è possibile conteggiare le combinazioni per ogni esito e calcolare la probabilità corrispondente
EsitoNumero
comb.		Prob.
21+1     1\displaystyle \frac{1}{36} = 2,77.. %
31+22+1    2\displaystyle \frac{2}{36}= \displaystyle \frac{1}{18}= 5,55.. %
41+32+23+1   3\displaystyle \frac{3}{36}= \displaystyle \frac{1}{12}= 8,33.. %
51+42+33+24+1  4\displaystyle \frac{4}{36}= \displaystyle \frac{1}{9}= 11,11.. %
61+52+43+34+25+1 5\displaystyle \frac{5}{36} = 13,88.. %
71+62+53+44+35+26+16\displaystyle \frac{6}{36}= \displaystyle \frac{1}{6}= 16,66.. %
8 2+63+54+45+36+25\displaystyle \frac{5}{36} = 13,88.. %
9  3+64+55+46+34\displaystyle \frac{4}{36}= \displaystyle \frac{1}{9}= 11,11.. %
10   4+65+56+43\displaystyle \frac{3}{36}= \displaystyle \frac{1}{12}= 8,33.. %
11    5+66+52\displaystyle \frac{2}{36}= \displaystyle \frac{1}{18}= 5,55.. %
12     6+61\displaystyle \frac{1}{36} = 2,77.. %
361

Sia X la variabile casuale “somma dei punti realizzati lanciando due dadi”, allora

\displaystyle x_i\displaystyle p_i\displaystyle p_i\cdot x_i\displaystyle x_i^2\displaystyle p_i\cdot x_i^2\displaystyle |x_i-m|\displaystyle p_i|x_i-m|\displaystyle (x_i-m)^2\displaystyle p_i(x_i-m)^2
2\displaystyle \frac{1}{36}\displaystyle \frac{1}{18}4\displaystyle \frac{1}{9}5\displaystyle \frac{5}{36}25\displaystyle \frac{25}{36}
3\displaystyle \frac{1}{18}\displaystyle \frac{1}{6}9\displaystyle \frac{1}{2}4\displaystyle \frac{2}{9}16\displaystyle \frac{8}{9}
4\displaystyle \frac{1}{12}\displaystyle \frac{1}{3}16\displaystyle \frac{4}{3}3\displaystyle \frac{1}{4}9\displaystyle \frac{3}{4}
5\displaystyle \frac{1}{9}\displaystyle \frac{5}{9}25\displaystyle \frac{25}{9}2\displaystyle \frac{2}{9}4\displaystyle \frac{4}{9}
6\displaystyle \frac{5}{36}\displaystyle \frac{5}{6}3651\displaystyle \frac{5}{36}1\displaystyle \frac{5}{36}
7\displaystyle \frac{1}{6}\displaystyle \frac{7}{6}49\displaystyle \frac{49}{6}0000
8\displaystyle \frac{5}{36}\displaystyle \frac{10}{9}64\displaystyle \frac{80}{6}1\displaystyle \frac{5}{36}1\displaystyle \frac{5}{36}
9\displaystyle \frac{1}{9}18192\displaystyle \frac{2}{9}4\displaystyle \frac{4}{9}
10\displaystyle \frac{1}{12}\displaystyle \frac{5}{6}100\displaystyle \frac{25}{3}3\displaystyle \frac{1}{4}9\displaystyle \frac{3}{4}
11\displaystyle \frac{1}{18}\displaystyle \frac{11}{18}121\displaystyle \frac{121}{18}4\displaystyle \frac{2}{9}16\displaystyle \frac{8}{9}
12\displaystyle \frac{1}{36}\displaystyle \frac{1}{3}14445\displaystyle \frac{5}{36}25\displaystyle \frac{25}{36}
  7 \displaystyle \frac{329}{6} \displaystyle \frac{35}{18}110\displaystyle \frac{35}{6}
  \displaystyle M(X) \displaystyle M(X^2) \displaystyle \delta(X)\displaystyle dev(X)\displaystyle var(X)

Osserva

Media\displaystyle M(X)\displaystyle \sum_i p_i\cdot x_i= \displaystyle 7
Scarto medio assoluto\displaystyle \delta(X)\displaystyle \sum _{i}p_i|x_{i}-m|}= \displaystyle \frac{35}{18}= 1,9444
Devianza\displaystyle dev(x)\displaystyle \sum_{i}(x_i-m)^2= \displaystyle 110
Varianza\displaystyle var(X)\displaystyle \sum_i p_i(x_i-m)^2= \displaystyle \frac{35}{6}= 5,8333…
Deviazione standard\sigma (X)\displaystyle \sqrt{\sum_i p_i(x_i-m)^2}= \displaystyle \sqrt{\frac{35}{6}}= 2,4152…
Deviazione standard relativa\displaystyle \sigma^{*}(X)\displaystyle \frac{\sigma(X)}{|M(X)|}= \displaystyle \sqrt{\frac{35}{6}} \cdot \frac{1}{7}= 0,345…
…\displaystyle M(X^2)\displaystyle \sum_ip_i\cdot x_i^2= \displaystyle \frac{329}{6}= 54,8333
Varianzavar(X)\displaystyle M(X^2)-[M(X)]^2= \displaystyle \frac{329}{6}-\left(7\right)^2= 5,8333…

Premio equo?

EsitoProbabilitàPremio equo
2\displaystyle \frac{1}{36} \displaystyle \frac{36}{1}36,0
3\displaystyle \frac{1}{18}\displaystyle \frac{18}{1}18,0
4\displaystyle \frac{1}{12}\displaystyle \frac{12}{1}12,0
5\displaystyle \frac{1}{9}\displaystyle \frac{9}{1}9,0
6 \displaystyle \frac{5}{36}\displaystyle \frac{36}{5}7,2
7\displaystyle \frac{1}{6}\displaystyle \frac{6}{1}6,0
8 \displaystyle \frac{5}{36}\displaystyle \frac{36}{5}7,2
9\displaystyle \frac{1}{9}\displaystyle \frac{9}{1}9,0
10\displaystyle \frac{1}{12}\displaystyle \frac{12}{1}12,0
11\displaystyle \frac{1}{18}\displaystyle \frac{18}{1}18,0
12 \displaystyle \frac{1}{36}\displaystyle \frac{36}{1}36,0
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