Theory of Stochastic Signals/Binomial Distribution - Revision history

Guenter at 16:32, 7 February 2024

2024-02-07T16:32:23Z

Show changes

Guenter at 14:28, 13 December 2022

2022-12-13T14:28:11Z

Hwang at 16:46, 12 December 2022

2022-12-12T16:46:36Z

Hwang at 16:44, 12 December 2022

2022-12-12T16:44:22Z

Hwang at 16:32, 12 December 2022

2022-12-12T16:32:39Z

Noah at 18:48, 19 April 2022

2022-04-19T18:48:42Z

Noah at 18:48, 19 April 2022

2022-04-19T18:48:21Z

Noah at 18:38, 19 April 2022

2022-04-19T18:38:15Z

Noah at 16:33, 15 February 2022

2022-02-15T16:33:55Z

Bene: Text replacement - "root mean square" to "standard deviation"

2022-02-15T15:50:30Z

Text replacement - "root mean square" to "standard deviation"

@@ Line 87: / Line 87: @@
-Use the interactive HTML5/JS applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson distribution"]]&nbsp; to find the binomial probabilities for any&nbsp; $I$&nbsp; and&nbsp; $p$&nbsp;.
+&rArr; &nbsp; Use the interactive HTML5/JS applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson distribution"]]&nbsp; to find the binomial probabilities for any&nbsp; $I$&nbsp; and&nbsp; $p$&nbsp;.
@@ Line 105: / Line 105: @@
 *the first order moment &nbsp; &rArr; &nbsp; "linear mean":
 :$$m_1 ={\rm E}\big[z\big]= I\cdot p,$$
-*the second order moment &nbsp; &rArr; &nbsp; "quadratic mean":
+*the second order moment:
 :$$m_2 ={\rm E}\big[z^2\big]= (I^2-I)\cdot p^2+I\cdot p,$$
 *the variance by applying "Steiner's theorem":

@@ Line 61: / Line 61: @@
 If one transmits blocks each of&nbsp; $I =10$&nbsp; binary symbols over a channel
 *with probability&nbsp; $p = 0.01$&nbsp; that one symbol is falsified &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 1$,&nbsp; and
-*correspondingly with probability&nbsp; $1 - p = 0.99$&nbsp; for a unfalsified  symbol &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 0$,
+*correspondingly with probability&nbsp; $1 - p = 0.99$&nbsp; for an unfalsified  symbol &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 0$,

@@ Line 60: / Line 60: @@
 If one transmits blocks each of&nbsp; $I =10$&nbsp; binary symbols over a channel
-*with probability&nbsp; $p = 0.01$&nbsp; that one symbol is corrupted &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 1$,&nbsp; and
+*with probability&nbsp; $p = 0.01$&nbsp; that one symbol is falsified &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 1$,&nbsp; and
-*correspondingly with probability&nbsp; $1 - p = 0.99$&nbsp; for a uncorrupted  symbol &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 0$,
+*correspondingly with probability&nbsp; $1 - p = 0.99$&nbsp; for a unfalsified  symbol &nbsp; &rArr; &nbsp; random variable&nbsp; $e_i = 0$,
@@ Line 67: / Line 67: @@
 :$$f=\sum_{i=1}^{I}e_i.$$
-This random variable&nbsp; $f$&nbsp; can take all integer values between&nbsp; $0$&nbsp; (no symbol is corrupted)&nbsp; and&nbsp; $I$&nbsp; (all symbols symbol are corrupted)&nbsp;.&nbsp;
+This random variable&nbsp; $f$&nbsp; can take all integer values between&nbsp; $0$&nbsp; (no symbol is falsified)&nbsp; and&nbsp; $I$&nbsp; (all symbols symbol are falsified)&nbsp;.&nbsp;
-*We denote the probabilities for&nbsp; $\mu$&nbsp; corruptions by&nbsp; $p_μ$.
+*We denote the probabilities for&nbsp; $\mu$&nbsp; falsifications by&nbsp; $p_μ$.
 *The case where all&nbsp; $I$&nbsp; symbols are correctly transmitted occurs with probability&nbsp; $p_0 = 0.99^{10} ≈ 0.9044$&nbsp;.
 *This also follows from the binomial formula for&nbsp; $μ = 0$&nbsp; considering the definition&nbsp; $10\, \text{ over }\, 0 = 1$.
@@ Line 74: / Line 74: @@
 :$$p_1 = \rm 10\cdot 0.01\cdot 0.99^9\approx 0.0914.$$
 ::The first factor considers that there are exactly&nbsp; $10\, \text{ over }\, 1 = 10$&nbsp; possibilities for the position of a single error.&nbsp;
-::The other two factors take into account that one symbol must be corrupted and nine must be transmitted correctly if&nbsp; $f =1$&nbsp; is to hold.
+::The other two factors take into account that one symbol must be falsified and nine must be transmitted correctly if&nbsp; $f =1$&nbsp; is to hold.
 *For&nbsp; $f =2$&nbsp; there are clearly more combinations,&nbsp; namely&nbsp; $10\, \text{ over }\, 2 = 45$, &nbsp; and we get
 :$$p_2 = \rm 45\cdot 0.01^2\cdot 0.99^8\approx 0.0041.$$
@@ Line 122: / Line 122: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
-As in&nbsp; $\text{Example 3}$,&nbsp; we consider a block of&nbsp; $I =10$&nbsp; binary symbols,&nbsp; each of which is independently corrupted with probability&nbsp; $p = 0.01$&nbsp;.&nbsp;
+As in&nbsp; $\text{Example 3}$,&nbsp; we consider a block of&nbsp; $I =10$&nbsp; binary symbols,&nbsp; each of which is independently falsified with probability&nbsp; $p = 0.01$&nbsp;.&nbsp;
 Then holds:
@@ Line 129: / Line 129: @@
-In contrast,&nbsp; in the completely corrupted channel &nbsp;   ⇒  &nbsp; bit error probability&nbsp; $p = 1/2$&nbsp; results in the values
+In contrast,&nbsp; in the completely falsified channel &nbsp;   ⇒  &nbsp; bit error probability&nbsp; $p = 1/2$&nbsp; results in the values
 *$m_f  = 5$ &nbsp;   ⇒  &nbsp; on average,&nbsp; five of the ten bits within a block are wrong,

@@ Line 9: / Line 9: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The&nbsp; '''binomial distribution'''&nbsp; represents an important special case for the occurrence probabilities of a discrete random variable.
+The&nbsp; &raquo;'''binomial distribution'''&laquo;&nbsp; represents an important special case for the occurrence probabilities of a discrete random variable.
 To derive the binomial distribution,&nbsp; we assume that&nbsp; $I$&nbsp; binary and statistically independent random variables&nbsp; $b_i$&nbsp; each can achieve
@@ Line 33: / Line 33: @@
 {{BlaueBox|TEXT=
 $\text{Calculation rule:}$&nbsp;
-For the&nbsp;  '''probabilities of the binomial distribution'''&nbsp;  with&nbsp;  $μ = 0, \hspace{0.1cm}\text{...} \hspace{0.1cm}, \ I$:
+For the&nbsp; &raquo;'''probabilities of the binomial distribution'''&laquo;&nbsp;  with&nbsp;  $μ = 0, \hspace{0.1cm}\text{...} \hspace{0.1cm}, \ I$:
 :$$p_\mu = {\rm Pr}(z=\mu)={I \choose \mu}\cdot p\hspace{0.05cm}^\mu\cdot ({\rm 1}-p)\hspace{0.05cm}^{I-\mu}.$$
 *The first term here indicates the number of combinations &nbsp; $($read:&nbsp;  $I\ \text{  over  }\ μ)$:
@@ Line 40: / Line 40: @@
 Additional notes:
-*For very large values of&nbsp;  $I$,&nbsp; the binomial distribution can be approximated by the&nbsp;  [[Theory_of_Stochastic_Signals/Poisson_Distribution|Poisson distribution]]&nbsp; described in the next section.
+*For very large values of&nbsp;  $I$,&nbsp; the binomial distribution can be approximated by the&nbsp;  [[Theory_of_Stochastic_Signals/Poisson_Distribution|$\text{Poisson distribution}$]]&nbsp; described in the next section.
-*If at the same time the product&nbsp;  $I · p \gg 1$,&nbsp; then according to&nbsp;  [https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem de Moivre–Laplace's (central limit) theorem],&nbsp; the Poisson distribution&nbsp;  (and hence the binomial distribution)&nbsp; transitions to a discrete&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distribution]].
+*If at the same time the product&nbsp;  $I · p \gg 1$,&nbsp; then according to&nbsp;  [https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem $\text{de Moivre–Laplace's (central limit) theorem}$],&nbsp; the Poisson distribution&nbsp;  (and hence the binomial distribution)&nbsp; transitions to a discrete&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|$\text{Gaussian distribution}$]].
@@ Line 87: / Line 87: @@
-Use the interactive HTML5/JS applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|Binomial and Poisson distribution]]&nbsp; to find the binomial probabilities for any&nbsp; $I$&nbsp; and&nbsp; $p$&nbsp;.
+Use the interactive HTML5/JS applet&nbsp; [[Applets:Binomial_and_Poisson_Distribution_(Applet)|"Binomial and Poisson distribution"]]&nbsp; to find the binomial probabilities for any&nbsp; $I$&nbsp; and&nbsp; $p$&nbsp;.
@@ Line 93: / Line 93: @@
 <br>
 You can calculate the moments in general using the equations in the chapters&nbsp;
-# [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|Moments of a Discrete Random Variable]],
+# [[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable|"Moments of a Discrete Random Variable"]],
-# [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution|Probabilities of the Binomial Distribution]].
+# [[Theory_of_Stochastic_Signals/Binomial_Distribution#Probabilities_of_the_binomial_distribution|"Probabilities of the Binomial Distribution"]].
@@ Line 100: / Line 100: @@
 $\text{Calculation rules:} $&nbsp;
-For the&nbsp; '''$k$-th order moment'''&nbsp; of a binomially distributed random variable,&nbsp; the general rule is:
+For the&nbsp; &raquo;'''$k$-th order moment'''&laquo;&nbsp; of a binomially distributed random variable,&nbsp; the general rule is:
 :$$m_k={\rm E}\big[z^k\big]=\sum_{\mu={\rm 0} }^{I}\mu^k\cdot{I \choose \mu}\cdot p\hspace{0.05cm}^\mu\cdot ({\rm 1}-p)\hspace{0.05cm}^{I-\mu}.$$
 From this,&nbsp; we obtain for after some transformations

@@ Line 1: / Line 1: @@
 {{Header
-|Untermenü=Dicrete Random Variable
+|Untermenü=Discrete Random Variable
 |Vorherige Seite=Moments of a Discrete Random Variable
 |Nächste Seite=Poissonverteilung

@@ Line 107: / Line 107: @@
 *the second order moment &nbsp; &rArr; &nbsp; "quadratic mean":
 :$$m_2 ={\rm E}\big[z^2\big]= (I^2-I)\cdot p^2+I\cdot p,$$
-*the variance by applyings "Steiner's theorem":
+*the variance by applying "Steiner's theorem":
 :$$\sigma^2 = {m_2-m_1^2} = {I \cdot p\cdot (1-p)} ,$$
 *the standard deviation:

@@ Line 47: / Line 47: @@
 $\text{Example 2:}$&nbsp;
 The graph shows the probabilities of the binomial distribution are for&nbsp; $I =6$&nbsp;  and&nbsp; $p =0.4$.&nbsp;
-[[File:P_ID203__Sto_T_2_3_S2_neu.png |frame|Binomial distribution probabilites]]
+[[File:P_ID203__Sto_T_2_3_S2_neu.png |frame|Binomial distribution probabilities]]
 *Thus&nbsp; $M = I+1=7$&nbsp; probabilities are different from zero.

@@ Line 109: / Line 109: @@
 *the variance by applyings "Steiner's theorem":
 :$$\sigma^2 = {m_2-m_1^2} = {I \cdot p\cdot (1-p)} ,$$
-*the root mean square value&nbsp; $\rm (rms)$&nbsp; or&nbsp;  "standard deviation":
+*the standard deviation value&nbsp; $\rm (rms)$&nbsp; or&nbsp;  "standard deviation":
 :$$\sigma =  \sqrt{I \cdot p\cdot (1-p)}.$$}}