Analisi di QUATTRO dadi

di
EsitoNumero
comb.		Prob.
41+1+1+1     1\displaystyle \frac{1}{1296}= 0,077… %
51+1+1+2
1+1+2+1
1+2+1+1		2+1+1+1    4\displaystyle \frac{1}{324}= 0,3… %
61+1+1+3
1+1+2+2
1+1+3+1
1+2+1+2
1+2+2+1
1+3+1+1		2+1+1+2
2+1+2+1
2+2+1+1		3+1+1+1   10\displaystyle \frac{5}{648}= 0,77… %
71+1+1+4
…		2+1+1+3
…		3+1+1+2
…		4+1+1+1  20\displaystyle \frac{5}{324}= 1,54… %
81+1+1+5
…		2+1+1+4
…		3+1+1+3
…		4+1+1+2
…		5+1+1+1 35\displaystyle \frac{35}{1296}= 2,7… %
91+1+1+6
…		2+1+1+5
…		3+1+1+4
…		4+1+1+3
…		5+1+1+2
…		6+1+1+156\displaystyle \frac{7}{162}= 4,32… %
101+1+2+6
…		2+1+1+6
…		3+1+1+5
…		4+1+1+4
…		5+1+1+3
…		6+1+1+2
…		80\displaystyle \frac{5}{81}= 6,17… %
111+1+3+6
…		2+1+2+6
…		3+1+1+6
…		4+1+1+5
…		5+1+1+4
…		6+1+1+3
…		104\displaystyle \frac{13}{162}= 8,0… %
121+1+4+6
…		2+1+3+6
…		3+1+2+6
…		4+1+1+6
…		5+1+1+5
…		6+1+1+4
…		125\displaystyle \frac{125}{1296}= 9,6… %
131+1+5+6
…		2+1+4+6
…		3+1+3+6
…		4+1+2+6
…		5+1+1+6
…		6+1+1+5
…		140\displaystyle \frac{35}{324}= 10,8… %
141+1+6+6
…		2+1+5+6
…		3+1+4+6
…		4+1+3+6
…		5+1+2+6
…		6+1+1+6
…		146\displaystyle \frac{73}{648}= 11,… %
151+2+6+6
…		2+1+6+6
…		3+1+5+6
…		4+1+4+6
…		5+1+3+6
…		6+1+2+6
…		140\displaystyle \frac{35}{324}= 10,8… %
161+3+6+6
…		2+2+6+6
…		3+1+6+6
…		4+1+5+6
…		5+1+4+6
…		6+1+3+6
…		125\displaystyle \frac{125}{1296}= 9,6… %
171+4+6+6
…		2+3+6+6
…		3+2+6+6
…		4+1+6+6
…		5+1+5+6
…		6+1+4+6
…		104\displaystyle \frac{13}{162}= 8,0… %
181+5+6+6
…		2+4+6+6
…		3+3+6+6
…		4+2+6+6
…		5+1+6+6
…		6+1+5+6
…		80\displaystyle \frac{5}{81}= 6,17… %
191+6+6+62+5+6+6
…		3+4+6+6
…		4+3+6+6
…		5+2+6+6
…		6+1+6+6
…		56\displaystyle \frac{7}{162}= 4,32… %
20 2+6+6+63+5+6+6
…		4+4+6+6
…		5+3+6+6
…		6+2+6+6
…		35\displaystyle \frac{35}{1296}= 2,7… %
21  3+6+6+64+5+6+6
…		5+4+6+6
…		6+3+6+6
…		20\displaystyle \frac{5}{324}= 1,54… %
22   4+6+6+65+5+6+6
…		6+4+6+6
…		10\displaystyle \frac{5}{648}= 0,77… %
23    5+6+6+66+5+6+6
6+6+5+6
6+6+6+5		4\displaystyle \frac{1}{324}= 0,3… %
24     6+6+6+61\displaystyle \frac{1}{1296}= 0,077… %

Sia X la variabile casuale “somma dei punti realizzati lanciando quattro 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
4\displaystyle \frac{1}{1296}\displaystyle \frac{1}{324}16\displaystyle \frac{1}{81}10\displaystyle \frac{5}{648}100\displaystyle \frac{25}{324}
5\displaystyle \frac{1}{324}\displaystyle \frac{5}{324}25\displaystyle \frac{25}{424}9\displaystyle \frac{1}{36}81\displaystyle \frac{1}{4}
6\displaystyle \frac{5}{648}\displaystyle \frac{5}{108}36\displaystyle \frac{5}{18}8\displaystyle \frac{5}{81}64\displaystyle \frac{40}{81}
7\displaystyle \frac{5}{324}\displaystyle \frac{35}{324}49\displaystyle \frac{245}{324}7\displaystyle \frac{35}{324}49\displaystyle \frac{245}{324}
8\displaystyle \frac{35}{1296}\displaystyle \frac{35}{162}64\displaystyle \frac{140}{81}6\displaystyle \frac{35}{216}36\displaystyle \frac{35}{36}
9\displaystyle \frac{7}{162}\displaystyle \frac{7}{18}81\displaystyle \frac{7}{2}5\displaystyle \frac{35}{162}25\displaystyle \frac{175}{162}
10\displaystyle \frac{5}{81}\displaystyle \frac{50}{81}100\displaystyle \frac{500}{81}4\displaystyle \frac{20}{81}16\displaystyle \frac{80}{81}
11\displaystyle \frac{13}{162}\displaystyle \frac{143}{162}121\displaystyle \frac{1573}{162}3\displaystyle \frac{13}{54}9\displaystyle \frac{13}{18}
12\displaystyle \frac{125}{1296}\displaystyle \frac{125}{108}144\displaystyle \frac{125}{9}2\displaystyle \frac{125}{648}4\displaystyle \frac{125}{324}
13\displaystyle \frac{35}{324}\displaystyle \frac{455}{324}169\displaystyle \frac{5915}{324}1\displaystyle \frac{35}{324}1\displaystyle \frac{35}{324}
14\displaystyle \frac{73}{648}\displaystyle \frac{511}{324}196\displaystyle \frac{3757}{162}0000
15\displaystyle \frac{35}{324}\displaystyle \frac{175}{108}225\displaystyle \frac{875}{36}1\displaystyle \frac{35}{324}1\displaystyle \frac{35}{324}
16\displaystyle \frac{125}{1296}\displaystyle \frac{125}{81}256\displaystyle \frac{2000}{81}2\displaystyle \frac{125}{648}4\displaystyle \frac{125}{324}
17\displaystyle \frac{13}{162}\displaystyle \frac{221}{162}289\displaystyle \frac{3757}{162}3\displaystyle \frac{13}{54}9\displaystyle \frac{13}{18}
18\displaystyle \frac{5}{81}\displaystyle \frac{10}{9}324\displaystyle \frac{20}{1}4\displaystyle \frac{20}{81}16\displaystyle \frac{80}{81}
19\displaystyle \frac{7}{162}\displaystyle \frac{133}{162}381\displaystyle \frac{2527}{162}5\displaystyle \frac{35}{162}25\displaystyle \frac{175}{162}
20\displaystyle \frac{35}{1296}\displaystyle \frac{175}{324}400\displaystyle \frac{875}{81}6\displaystyle \frac{35}{216}36\displaystyle \frac{35}{36}
21\displaystyle \frac{5}{324}\displaystyle \frac{35}{108}421\displaystyle \frac{245}{36}7\displaystyle \frac{35}{324}49\displaystyle \frac{245}{324}
22\displaystyle \frac{5}{648}\displaystyle \frac{55}{324}484\displaystyle \frac{605}{162}8\displaystyle \frac{5}{81}64\displaystyle \frac{40}{81}
23\displaystyle \frac{1}{324}\displaystyle \frac{23}{324}529\displaystyle \frac{529}{324}9\displaystyle \frac{1}{36}81\displaystyle \frac{1}{4}
24\displaystyle \frac{1}{1296}\displaystyle \frac{1}{54}576\displaystyle \frac{4}{9}10\displaystyle \frac{5}{648}100\displaystyle \frac{25}{324}
 1\displaystyle 14 \displaystyle \frac{623}{3} \displaystyle \frac{889}{324}770\displaystyle \frac{35}{3}
  \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= 14
Scarto medio assoluto\displaystyle \delta(X)= \displaystyle \sum _{i}p_i|x_{i}-m|}= \displaystyle \frac{889}{324}= 2.743827…
Devianza\displaystyle dev(x)= \displaystyle \sum_{i}(x_i-m)^2= 770
Varianza\displaystyle var(X)= \displaystyle \sum_i p_i(x_i-m)^2= \displaystyle \frac{35}{3}= 11,66…
Deviazione standard\sigma (X)= \displaystyle \sqrt{\sum_i p_i(x_i-m)^2}= \displaystyle \frac{\sqrt{105}}{3}}= 3.41565…
Deviazione standard relativa\displaystyle \sigma^{*}(X)= \displaystyle \frac{\sigma(X)}{|M(X)|}= \displaystyle \frac{\sqrt{105}}{42}}= 0.243975…
…\displaystyle M(X^2)= \displaystyle \sum_ip_i\cdot x_i^2= \displaystyle \frac{623}{3}= 207,66…
Varianzavar(X)= \displaystyle M(X^2)-[M(X)]^2= \displaystyle \frac{623}{3}-(14)^2 = \displaystyle \frac{35}{3}= 3.41565…