Estatística Aplicada - Aula 08

Exercícios sobre Distribuição Polinomial - Aula passada

Gabriel Coelho Soares

1)

$$ P = 0,5 $$

$$ Q = 0,5 $$

$$ n = 20 $$

$$ k = 8 $$

$$ P = C_{20.8} \times 0.5^8 \times 0.5^{12} $$

$$ \begin{align} P = 125970 \cdot 0.00390625 \\ P = 0.1201343413 \cdot 100 \\ \color{red} P = 12.01 \end{align} $$

2) a)

$P = 0.4$

$Q = 0.6$

$n = 25$

$k = 2$

$$ \begin{align} P = C_{25.2} \cdot 0.4^2 \cdot 0.6^{23} \\ P = 300 \cdot 0.16 \cdot 0.0000078973 \\ P = 0.0003790 \cdot 100 \\ \color{red} P = 0.0379% \end{align} $$

b)

$P = 0.4$

$Q = 0.6$

$n = 25$

$k \geq 3 = 1-[p(x=0) + p(x=1) + p(x=2)]$

p(x=0)

$$ \begin{align} C_{25} \cdot 0.4^0 \cdot 0.6^{25} \\ 1 \cdot 1 \cdot \color{red}0.000002843 \end{align} $$

p(x=1)

$$ \begin{align} C_{25.1} \cdot 0.4^1 \cdot 0.6^{24} \\ 25 \cdot 0.4 \cdot 0.000004738 \\ \color{red}0.000047380 \end{align} $$

p(x=2)

$$ \begin{align} C_{25.2} \cdot 0.4^2 \cdot 0.6^{23} \\ 300 \cdot 0.16 \cdot 0.000007897 \\ \color{red}0.000379056 \end{align} $$

$$ \begin{align} 1 - \sum_{0}^2p \\ 1 - 0.000429279 = 0.9995 \\ \color{red}99.95% \end{align} $$

c)

$P = 0.4$

$Q = 0.6$

$n = 25$

$k \geq 2 = 1-[p(x=0) + p(x=1)]$

p(x=0)

$$ \begin{align} C_{6} \cdot 0.6^0 \cdot 0.4^6 \\ 1 \cdot 1 \cdot 0.00096 \end{align} $$

p(x=1)

$$ \begin{align} C_{6.1} \cdot 0.6^1 \cdot 0.4^5 \\ 6 \cdot 0.6 \cdot 0.01024 = 0.0368640 \\ \\ \\ \\ => 0.040960 \cdot 100 \\ \color{red} 4.096% \end{align} $$

3) a)

$p=0.18$

$q=0.82$

$n=10$

$k \geq 2 = 1-\sum_0^1p$

p(x=0)

$$ \begin{align} C_{10} \cdot 0.18^0 \cdot 0.82^{10} \\ 1 \cdot 1 \cdot 0.137448 \end{align} $$

p(x=1)

$$ \begin{align} C_{10.1} \cdot 0.18^1 \cdot 0.82^9 \\ 10 \cdot 0.18 \cdot 0.167619 = 0.301415 \\ \\ \\ 1 - 0.4391 = 0.5609 \\ \color{red} 56.09% \end{align} $$

b)

$$ \begin{align} k=0 = 0.137448 \\ k = 1 = 0.301415 \\ k = 2 = ? \\ \\ \\ C_{10.2} \cdot 0.18^2 \cdot 0.82^8 \\ 45 \cdot 0.03240 \cdot 0.2044 = 0.29803 \\ \\ \\ k_1 + k_2 + k_3 = 0.7368 = \color{red} 73.68% \end{align} $$

4)

$$ \begin{align} p=0.18 \\ q=0.82\\ n=10\\ k \leq 9 \\ \\ k=3 \\ C_{10.3} \cdot 0.18^3 \cdot 0.82^7 \\ 120 \cdot0.005832 \cdot 0.24928 = 0.174459 \\ \\ k=4 \\ C_{10.4} \cdot 0.18^4 \cdot 0.82^6 \\ 210 \cdot0.001049 \cdot 0.0.3040 = 0.06696 \\ \\ k=5 \\ C_{10.5} \cdot 0.18^5 \cdot 0.82^5 \\ 252 \cdot0.0001889 \cdot 0.3707 = 0.017648 \\ \\ k=6 \\ C_{10.6} \cdot 0.18^6 \cdot 0.82^4 \\ 210 \cdot0.0000340 \cdot 0.45212 = 0.0032281 \\ \\ k=7 \\ C_{10.7} \cdot 0.18^7 \cdot 0.82^3 \\ 120 \cdot0.0000061 \cdot 0.5513 = 0.000403601 \\ \\ k=8 \\ C_{10.8} \cdot 0.18^8 \cdot 0.82^2 \\ 45 \cdot0.000001122 \cdot 0.6724 = 0.000033344 \\ \\ k=9 \\ C_{10.9} \cdot 0.18^9 \cdot 0.82^1 \\ 10 \cdot0.000000198 \cdot 0.82 = 0.000001624 \\ \\ \\ \\ \\ p = \sum_0^9k = 0.9959 \\ \color{red} = 99.59% \end{align} $$