In Class Exercise: Binomial Distribution & Error Rates
putt
12/1/2018
1 Binomial (n=20, \(\pi\)) Distribution
1.1 Probability Mass Function
Here we consider a random variable, X, with a Binomial (n=20, \(\pi\)) distribution. The table below shows the probability mass function, specifically the possible values of the random variable and their probabilities. For example the \(P(X=3|\pi=.25)=.134\).
| x | pi.eq.25 | pi.eq.5 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|
| 0 | 0.003 | 0.000 | 0.000 | 0.000 |
| 1 | 0.021 | 0.000 | 0.000 | 0.000 |
| 2 | 0.067 | 0.000 | 0.000 | 0.000 |
| 3 | 0.134 | 0.001 | 0.000 | 0.000 |
| 4 | 0.190 | 0.005 | 0.000 | 0.000 |
| 5 | 0.202 | 0.015 | 0.000 | 0.000 |
| 6 | 0.169 | 0.037 | 0.000 | 0.000 |
| 7 | 0.112 | 0.074 | 0.000 | 0.000 |
| 8 | 0.061 | 0.120 | 0.001 | 0.000 |
| 9 | 0.027 | 0.160 | 0.003 | 0.000 |
| 10 | 0.010 | 0.176 | 0.010 | 0.000 |
| 11 | 0.003 | 0.160 | 0.027 | 0.000 |
| 12 | 0.001 | 0.120 | 0.061 | 0.000 |
| 13 | 0.000 | 0.074 | 0.112 | 0.002 |
| 14 | 0.000 | 0.037 | 0.169 | 0.009 |
| 15 | 0.000 | 0.015 | 0.202 | 0.032 |
| 16 | 0.000 | 0.005 | 0.190 | 0.090 |
| 17 | 0.000 | 0.001 | 0.134 | 0.190 |
| 18 | 0.000 | 0.000 | 0.067 | 0.285 |
| 19 | 0.000 | 0.000 | 0.021 | 0.270 |
| 20 | 0.000 | 0.000 | 0.003 | 0.122 |
1.1.1 Graphs
Probability Mass Function for Binomial(n=20, pi) and pi betwen 0.25 and 0.9
1.2 Cumulative Distribution Functrion
Here we consider a random variable, X, with a Binomial (n=20, \(\pi\)) distrbiution. The table below shows the cumulative distribution function, specifically, x, the possible values of the random variable and \(P(X \leq x)\). For example the \(P(X \leq3|\pi=.25)=.225\).
| x | pi.eq.25 | pi.eq.5 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|
| 0 | 0.003 | 0.000 | 0.000 | 0.000 |
| 1 | 0.024 | 0.000 | 0.000 | 0.000 |
| 2 | 0.091 | 0.000 | 0.000 | 0.000 |
| 3 | 0.225 | 0.001 | 0.000 | 0.000 |
| 4 | 0.415 | 0.006 | 0.000 | 0.000 |
| 5 | 0.617 | 0.021 | 0.000 | 0.000 |
| 6 | 0.786 | 0.058 | 0.000 | 0.000 |
| 7 | 0.898 | 0.132 | 0.000 | 0.000 |
| 8 | 0.959 | 0.252 | 0.001 | 0.000 |
| 9 | 0.986 | 0.412 | 0.004 | 0.000 |
| 10 | 0.996 | 0.588 | 0.014 | 0.000 |
| 11 | 0.999 | 0.748 | 0.041 | 0.000 |
| 12 | 1.000 | 0.868 | 0.102 | 0.000 |
| 13 | 1.000 | 0.942 | 0.214 | 0.002 |
| 14 | 1.000 | 0.979 | 0.383 | 0.011 |
| 15 | 1.000 | 0.994 | 0.585 | 0.043 |
| 16 | 1.000 | 0.999 | 0.775 | 0.133 |
| 17 | 1.000 | 1.000 | 0.909 | 0.323 |
| 18 | 1.000 | 1.000 | 0.976 | 0.608 |
| 19 | 1.000 | 1.000 | 0.997 | 0.878 |
| 20 | 1.000 | 1.000 | 1.000 | 1.000 |
1.3 Binomial Distribution: One Minus Cumulative Distribution Functrion
Here we consider a random variable, X, with a Binomial (n=20, \(\pi\)) distrbiution. The table below shows one minus the cumulative distribution function, specifically, x, the possible values of the random variable and \(P(X\geq (x+1))=1-P(X \leq x)\).For example when \(\pi=.25\),the \(P(X \geq 4)=1-P(X\leq 3)=.775\).
| x | pi.eq.25 | pi.eq.5 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|
| 0 | 0.997 | 1.000 | 1.000 | 1.000 |
| 1 | 0.976 | 1.000 | 1.000 | 1.000 |
| 2 | 0.909 | 1.000 | 1.000 | 1.000 |
| 3 | 0.775 | 0.999 | 1.000 | 1.000 |
| 4 | 0.585 | 0.994 | 1.000 | 1.000 |
| 5 | 0.383 | 0.979 | 1.000 | 1.000 |
| 6 | 0.214 | 0.942 | 1.000 | 1.000 |
| 7 | 0.102 | 0.868 | 1.000 | 1.000 |
| 8 | 0.041 | 0.748 | 0.999 | 1.000 |
| 9 | 0.014 | 0.588 | 0.996 | 1.000 |
| 10 | 0.004 | 0.412 | 0.986 | 1.000 |
| 11 | 0.001 | 0.252 | 0.959 | 1.000 |
| 12 | 0.000 | 0.132 | 0.898 | 1.000 |
| 13 | 0.000 | 0.058 | 0.786 | 0.998 |
| 14 | 0.000 | 0.021 | 0.617 | 0.989 |
| 15 | 0.000 | 0.006 | 0.415 | 0.957 |
| 16 | 0.000 | 0.001 | 0.225 | 0.867 |
| 17 | 0.000 | 0.000 | 0.091 | 0.677 |
| 18 | 0.000 | 0.000 | 0.024 | 0.392 |
| 19 | 0.000 | 0.000 | 0.003 | 0.122 |
| 20 | 0.000 | 0.000 | 0.000 | 0.000 |
2 Binomial (n=40, \(\pi\)) Distribution
2.1 Probability Mass Function
Here we consider a random variable, X, with a Binomial (n=40, \(\pi\)) distrbiution. The table below shows the probability mass function, specifically the possible values of the random variable and their probabilities.
| x | pi.eq.25 | pi.eq.5 | pi.eq.65 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|---|
| 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 2 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 |
| 3 | 0.004 | 0.000 | 0.000 | 0.000 | 0.000 |
| 4 | 0.011 | 0.000 | 0.000 | 0.000 | 0.000 |
| 5 | 0.027 | 0.000 | 0.000 | 0.000 | 0.000 |
| 6 | 0.053 | 0.000 | 0.000 | 0.000 | 0.000 |
| 7 | 0.086 | 0.000 | 0.000 | 0.000 | 0.000 |
| 8 | 0.118 | 0.000 | 0.000 | 0.000 | 0.000 |
| 9 | 0.140 | 0.000 | 0.000 | 0.000 | 0.000 |
| 10 | 0.144 | 0.001 | 0.000 | 0.000 | 0.000 |
| 11 | 0.131 | 0.002 | 0.000 | 0.000 | 0.000 |
| 12 | 0.106 | 0.005 | 0.000 | 0.000 | 0.000 |
| 13 | 0.076 | 0.011 | 0.000 | 0.000 | 0.000 |
| 14 | 0.049 | 0.021 | 0.000 | 0.000 | 0.000 |
| 15 | 0.028 | 0.037 | 0.000 | 0.000 | 0.000 |
| 16 | 0.015 | 0.057 | 0.001 | 0.000 | 0.000 |
| 17 | 0.007 | 0.081 | 0.002 | 0.000 | 0.000 |
| 18 | 0.003 | 0.103 | 0.005 | 0.000 | 0.000 |
| 19 | 0.001 | 0.119 | 0.010 | 0.000 | 0.000 |
| 20 | 0.000 | 0.125 | 0.019 | 0.000 | 0.000 |
| 21 | 0.000 | 0.119 | 0.034 | 0.001 | 0.000 |
| 22 | 0.000 | 0.103 | 0.054 | 0.003 | 0.000 |
| 23 | 0.000 | 0.081 | 0.078 | 0.007 | 0.000 |
| 24 | 0.000 | 0.057 | 0.103 | 0.015 | 0.000 |
| 25 | 0.000 | 0.037 | 0.123 | 0.028 | 0.000 |
| 26 | 0.000 | 0.021 | 0.131 | 0.049 | 0.000 |
| 27 | 0.000 | 0.011 | 0.126 | 0.076 | 0.000 |
| 28 | 0.000 | 0.005 | 0.109 | 0.106 | 0.000 |
| 29 | 0.000 | 0.002 | 0.084 | 0.131 | 0.001 |
| 30 | 0.000 | 0.001 | 0.057 | 0.144 | 0.004 |
| 31 | 0.000 | 0.000 | 0.034 | 0.140 | 0.010 |
| 32 | 0.000 | 0.000 | 0.018 | 0.118 | 0.026 |
| 33 | 0.000 | 0.000 | 0.008 | 0.086 | 0.058 |
| 34 | 0.000 | 0.000 | 0.003 | 0.053 | 0.107 |
| 35 | 0.000 | 0.000 | 0.001 | 0.027 | 0.165 |
| 36 | 0.000 | 0.000 | 0.000 | 0.011 | 0.206 |
| 37 | 0.000 | 0.000 | 0.000 | 0.004 | 0.200 |
| 38 | 0.000 | 0.000 | 0.000 | 0.001 | 0.142 |
| 39 | 0.000 | 0.000 | 0.000 | 0.000 | 0.066 |
| 40 | 0.000 | 0.000 | 0.000 | 0.000 | 0.015 |
2.1.1 Graphs
Probability Mass Function for Binomial(n=40, pi) and pi betwen 0.25 and 0.9
2.2 Cumulative Distribution Function
Here we consider a random variable, X, with a Binomial (n=40, \(\pi\)) distrbiution. The table below shows the cumulative distribution function, specifically, x, the possible values of the random variable and \(P(X \leq x)\).
| x | pi.eq.25 | pi.eq.5 | pi.eq.65 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|---|
| 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 2 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 |
| 3 | 0.005 | 0.000 | 0.000 | 0.000 | 0.000 |
| 4 | 0.016 | 0.000 | 0.000 | 0.000 | 0.000 |
| 5 | 0.043 | 0.000 | 0.000 | 0.000 | 0.000 |
| 6 | 0.096 | 0.000 | 0.000 | 0.000 | 0.000 |
| 7 | 0.182 | 0.000 | 0.000 | 0.000 | 0.000 |
| 8 | 0.300 | 0.000 | 0.000 | 0.000 | 0.000 |
| 9 | 0.440 | 0.000 | 0.000 | 0.000 | 0.000 |
| 10 | 0.584 | 0.001 | 0.000 | 0.000 | 0.000 |
| 11 | 0.715 | 0.003 | 0.000 | 0.000 | 0.000 |
| 12 | 0.821 | 0.008 | 0.000 | 0.000 | 0.000 |
| 13 | 0.897 | 0.019 | 0.000 | 0.000 | 0.000 |
| 14 | 0.946 | 0.040 | 0.000 | 0.000 | 0.000 |
| 15 | 0.974 | 0.077 | 0.000 | 0.000 | 0.000 |
| 16 | 0.988 | 0.134 | 0.000 | 0.001 | 0.000 |
| 17 | 0.995 | 0.215 | 0.000 | 0.003 | 0.000 |
| 18 | 0.998 | 0.318 | 0.000 | 0.008 | 0.000 |
| 19 | 0.999 | 0.437 | 0.000 | 0.017 | 0.000 |
| 20 | 1.000 | 0.563 | 0.001 | 0.036 | 0.000 |
| 21 | 1.000 | 0.682 | 0.002 | 0.070 | 0.000 |
| 22 | 1.000 | 0.785 | 0.005 | 0.124 | 0.000 |
| 23 | 1.000 | 0.866 | 0.012 | 0.202 | 0.000 |
| 24 | 1.000 | 0.923 | 0.026 | 0.305 | 0.000 |
| 25 | 1.000 | 0.960 | 0.054 | 0.428 | 0.000 |
| 26 | 1.000 | 0.981 | 0.103 | 0.559 | 0.000 |
| 27 | 1.000 | 0.992 | 0.179 | 0.686 | 0.000 |
| 28 | 1.000 | 0.997 | 0.285 | 0.795 | 0.000 |
| 29 | 1.000 | 0.999 | 0.416 | 0.879 | 0.001 |
| 30 | 1.000 | 1.000 | 0.560 | 0.936 | 0.005 |
| 31 | 1.000 | 1.000 | 0.700 | 0.970 | 0.015 |
| 32 | 1.000 | 1.000 | 0.818 | 0.988 | 0.042 |
| 33 | 1.000 | 1.000 | 0.904 | 0.996 | 0.100 |
| 34 | 1.000 | 1.000 | 0.957 | 0.999 | 0.206 |
| 35 | 1.000 | 1.000 | 0.984 | 1.000 | 0.371 |
| 36 | 1.000 | 1.000 | 0.995 | 1.000 | 0.577 |
| 37 | 1.000 | 1.000 | 0.999 | 1.000 | 0.777 |
| 38 | 1.000 | 1.000 | 1.000 | 1.000 | 0.920 |
| 39 | 1.000 | 1.000 | 1.000 | 1.000 | 0.985 |
| 40 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
2.3 Binomial Distribution: One Minus Cumulative Distribution Functrion
Here we consider a random variable, X, with a Binomial (n=40, \(\pi\)) distrbiution. The table below shows one minus the cumulative distribution function, specifically, \(P(X > x)=P(X\geq x+1)\). For example when \(\pi=.25\), \(P(X \geq 6)=0.957\)
| x | pi.eq.25 | pi.eq.5 | pi.eq.65 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|---|
| 0 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 1 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 2 | 0.999 | 1.000 | 1.000 | 1.000 | 1.000 |
| 3 | 0.995 | 1.000 | 1.000 | 1.000 | 1.000 |
| 4 | 0.984 | 1.000 | 1.000 | 1.000 | 1.000 |
| 5 | 0.957 | 1.000 | 1.000 | 1.000 | 1.000 |
| 6 | 0.904 | 1.000 | 1.000 | 1.000 | 1.000 |
| 7 | 0.818 | 1.000 | 1.000 | 1.000 | 1.000 |
| 8 | 0.700 | 1.000 | 1.000 | 1.000 | 1.000 |
| 9 | 0.560 | 1.000 | 1.000 | 1.000 | 1.000 |
| 10 | 0.416 | 0.999 | 1.000 | 1.000 | 1.000 |
| 11 | 0.285 | 0.997 | 1.000 | 1.000 | 1.000 |
| 12 | 0.179 | 0.992 | 1.000 | 1.000 | 1.000 |
| 13 | 0.103 | 0.981 | 1.000 | 1.000 | 1.000 |
| 14 | 0.054 | 0.960 | 1.000 | 1.000 | 1.000 |
| 15 | 0.026 | 0.923 | 1.000 | 1.000 | 1.000 |
| 16 | 0.012 | 0.866 | 0.999 | 1.000 | 1.000 |
| 17 | 0.005 | 0.785 | 0.997 | 1.000 | 1.000 |
| 18 | 0.002 | 0.682 | 0.992 | 1.000 | 1.000 |
| 19 | 0.001 | 0.563 | 0.983 | 1.000 | 1.000 |
| 20 | 0.000 | 0.437 | 0.964 | 0.999 | 1.000 |
| 21 | 0.000 | 0.318 | 0.930 | 0.998 | 1.000 |
| 22 | 0.000 | 0.215 | 0.876 | 0.995 | 1.000 |
| 23 | 0.000 | 0.134 | 0.798 | 0.988 | 1.000 |
| 24 | 0.000 | 0.077 | 0.695 | 0.974 | 1.000 |
| 25 | 0.000 | 0.040 | 0.572 | 0.946 | 1.000 |
| 26 | 0.000 | 0.019 | 0.441 | 0.897 | 1.000 |
| 27 | 0.000 | 0.008 | 0.314 | 0.821 | 1.000 |
| 28 | 0.000 | 0.003 | 0.205 | 0.715 | 1.000 |
| 29 | 0.000 | 0.001 | 0.121 | 0.584 | 0.999 |
| 30 | 0.000 | 0.000 | 0.064 | 0.440 | 0.995 |
| 31 | 0.000 | 0.000 | 0.030 | 0.300 | 0.985 |
| 32 | 0.000 | 0.000 | 0.012 | 0.182 | 0.958 |
| 33 | 0.000 | 0.000 | 0.004 | 0.096 | 0.900 |
| 34 | 0.000 | 0.000 | 0.001 | 0.043 | 0.794 |
| 35 | 0.000 | 0.000 | 0.000 | 0.016 | 0.629 |
| 36 | 0.000 | 0.000 | 0.000 | 0.005 | 0.423 |
| 37 | 0.000 | 0.000 | 0.000 | 0.001 | 0.223 |
| 38 | 0.000 | 0.000 | 0.000 | 0.000 | 0.080 |
| 39 | 0.000 | 0.000 | 0.000 | 0.000 | 0.015 |
| 40 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
3 Binomial (n=8, \(\pi\)) Distribution
3.1 Probability Mass Function
Here we consider a random variable, X, with a Binomial (n=8, \(\pi\)) distrbiution. The table below shows the probability mass function, specifically the possible values of the random variable and their probabilities.
| x | pi.eq.25 | pi.eq.5 | pi.eq.75 | pi.eq.9 |
|---|---|---|---|---|
| 0 | 0.100 | 0.004 | 0.000 | 0.000 |
| 1 | 0.267 | 0.031 | 0.000 | 0.000 |
| 2 | 0.311 | 0.109 | 0.004 | 0.000 |
| 3 | 0.208 | 0.219 | 0.023 | 0.000 |
| 4 | 0.087 | 0.273 | 0.087 | 0.005 |
| 5 | 0.023 | 0.219 | 0.208 | 0.033 |
| 6 | 0.004 | 0.109 | 0.311 | 0.149 |
| 7 | 0.000 | 0.031 | 0.267 | 0.383 |
| 8 | 0.000 | 0.004 | 0.100 | 0.430 |
3.1.1 Graphs
Probability Mass Function for Binomial(n=8, pi) and pi betwen 0.25 and 0.9