Binomial Probability Distribution (2012)

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We first need to understand the Bernoulli Trials to learn about Binomial Probability Distribution.

Bernoulli Trials

Many experiments consist of repeated independent trials and each trial has only two possible outcomes such as head or tail, right or wrong, alive or dead, defective or non-defective, etc. If the probability of each outcome remains the same (constant) throughout the trials, then such trials are called the Bernoulli Trials.

Binomial Probability Distribution

Binomial Probability Distribution is a discrete probability distribution describing the results of an experiment known as the Bernoulli Process. The experiment having $n$ Bernoulli trials is called a Binomial Probability experiment possessing the following four conditions/ assumptions

  1. The experiment consists of $n$ repeated tasks.
  2. Each trial results in an outcome that may be classified as success or failure.
  3. The probability of success denoted by $p$ remains constant from trial to trial.
  4. The repeated trials are independent.

A Binomial trial can result in a success with probability $p$ and a failure with probability $1-p$ having $n-x$ number of failures, then the probability distribution of Binomial Random Variable, the number of successes in $n$ independent trial is:

\begin{align*}
P(X=x)&=\binom{n}{x} \, p^x \, q^{n-x} \\
&=\frac{n!}{x!(n-x)!}\, p^x \, q^{n-x}
\end{align*}

Binomial Probability Distribution

The Binomial probability distribution is the most widely used in situations of two outcomes. It was discovered by the Swiss mathematician Jakob Bernoulli (1654—1704) whose main work on “the ars Conjectandi” (the art of conjecturing) was published posthumously in Basel in 1713.

Mean of Binomial Distribution:   Mean = $\mu = np$

Variance of Binomial Distribution:  Variance = $npq$

Standard Deviation of Binomial Distribution:  Standard Deviation = $\sqrt{npq}$

Moment Coefficient of Skewness:

\begin{align*}
\beta_1 &= \frac{q-p}{\sqrt{npq}}  \\
&= \frac{1-2p}{\sqrt{npq}}
\end{align*}

Moment Coefficient of Kurtosis:  $\beta_3 = 3+\frac{1-6pq}{npq}$

Application of Binomial Probability Distribution

  • Quality control: In manufacturing, Binomial Probability Distribution can be used to determine the probability of finding a defective product in a batch.
  • Medical testing: It can be used to assess the probability of a specific number of positive test results in a group.
  • Opinion polls: Binomial Probability Distribution can be used to estimate the margin of error in a poll by considering the probability of getting a certain number of votes for a particular candidate.

By understanding the binomial distribution, you can analyze the probability of success in various scenarios with two possible outcomes.

FAQS about Binomial Probability Distribution

  1. What is a Binomial Experiment?
  2. Define Binomial Distribution?
  3. What are the important Assumptions of a Binomial experiment?
  4. What are the important applications of Binomial distribution?
  5. What are the characteristics of Binomial distribution?
  6. Write the probability distribution formula for a Binomial random variable.
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Generate Binomial Random Numbers in R Language

Coefficient of Determination: Model Selection (2012)

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$R^2$ pronounced R-squared (Coefficient of determination) is a useful statistic to check the regression fit value. $R^2$ measures the proportion of total variation about the mean $\bar{Y}$ explained by the regression. R is the correlation between $Y$ and $\hat{Y}$ and is usually the multiple correlation coefficient. The coefficient of determination ($R^2$) can take values as high as 1 or  (100%) when all the values are different i.e. $0\le R^2\le 1$.

Coefficient of Determination

When repeat runs exist in the data the value of $R^2$ cannot attain 1, no matter how well the model fits, because no model can explain the variation in the data due to the pure error. A perfect fit to data for which $\hat{Y}_i=Y_i$, $R^2=1$. If $\hat{Y}_i=\bar{Y}$, that is if $\beta_1=\beta_2=\cdots=\beta_{p-1}=0$ or if a model $Y=\beta_0 +\varepsilon$ alone has been fitted, then $R^2=0$. Therefore we can say that $R^2$ is a measure of the usefulness of the terms other than $\beta_0$ in the model.

Note that we must be sure that an improvement/ increase in $R^2$ value due to adding a new term (variable) to the model under study should have some real significance and is not because the number of parameters in the model is getting else to saturation point. If there is no pure error $R^2$ can be made unity.

\begin{align*}
R^2 &= \frac{\text {SS due to regression given}\, b_0}{\text{Total SS corrected for mean} \, \bar{Y}} \\
&= \frac{SS \, (b_1 | b_0)}{S_{YY}} \\
&= \frac{\sum(\hat{Y_i}-\bar{Y})^2} {\sum(Y_i-\bar{Y})^2}r \\
&= \frac{S^2_{XY}}{(S_{XY})(S_{YY})}
\end{align*}

where summation are over $i=1,2,\cdots, n$.

Coefficient of Determination
Coefficient of Determination

Interpreting R-Square $R^2$ does not indicate whether:

  • the independent variables (explanatory variables) are a cause of the changes in the dependent variable;
  • omitted-variable bias exists;
  • the correct regression was used;
  • the most appropriate set of explanatory variables has been selected;
  • there is collinearity (or multicollinearity) present in the data;
  • the model might be improved using transformed versions of the existing explanatory variables.

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What is Pseudo Random Process (2012)

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Pseudo Random Process

A pseudo random refers to a process that generates a sequence of numbers or events that appears random but actually, is not and is determined by a fixed set of rules. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process. Such a process is easier to produce than a genuinely random one and has the benefit that it can be used again and again to produce the same numbers and they are useful for testing and fixing software.

The generation of random numbers has many uses (mostly in Statistics, Random Sampling, and Simulation, Computer Modeling, Markov Chains, and Experimental Design). Before modern computing, researchers requiring random numbers would either generate them through various means like a coin, dice, cards, roulette wheels, card shuffling, etc., or use existing random number tables.

Pseudo Random Process

A pseudo-random variable is a variable that is created by a deterministic procedure (often a computer program or subroutine is used) which (generally) takes random bits as input. The pseudo random string will typically be longer than the original random string, but less random (less entropic, in the information-theory sense). This can be useful for randomized algorithms.

Pseudo-random numbers are computer-generated random numbers and they are not truly random because there is an inherent pattern in any sequence of pseudo numbers.

A question arises here why do we use something that is not truly random? The reasons behind the use of pseudo random process are:

  • Speed and Efficiency: Generating pseudo-random numbers is much faster and more efficient than using true random sources like physical processes.
  • Reproducibility: Using the same seed, one can reproduce the same sequence of pseudo-random numbers. which is useful for debugging or comparing simulations.

Read more about Random Number Process, Pseudo-Random Number Generation, and Linear Congruential Generator (LCG)

Read more about Pseudo-Random Number Generator

Generate Binomial Random Numbers in R

Linear Congruential Generator (LCG)

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A linear congruential generator (LCG) is an old algorithm that results in a sequence of pseudo-randomized numbers. Though, the algorithm of linear congruential generator is the oldest but best-known pseudorandom number generator method.

The building block of a simulation study is the ability to generate random numbers where a random number represents the value of a random variable uniformly distributed on (0,1).

The recurrence relation defines the generator:

\[X_{i+1}=(aX_i+C) \text{ Modulo } m\]

where $a$ and $m$ are given positive integers, $X_i$ is either $0,1, \dots, m-1$ and quantity $\frac{X_i}{m}$ is pseudo random number.

Conditions for Linear Congruential Generator

Some conditions are:

  1. $m>0$;  $m$ is usually large
  2. $0<a<m$;  ($a$ is the multiplier)
  3. $0\le c<m$ ($c$ is the increment)
  4. $0\le X_0 <m$ ($X_0$ is seed value or starting value)
  5. $c$ and $m$ are relatively prime numbers (there is no common factor between $c$ and $m$).
  6. $a-1$ is a multiple of every prime factor $m$
  7. $a-1$ is multiple of 4 if $m$ is multiple of 4
Linear Congruential Generator
Source: https://en.wikipedia.org/wiki/Linear_congruential_generator

“Two modulo-9 LCGs show how different parameters lead to different cycle lengths. Each row shows the state evolving until it repeats. The top row shows a generator with $m=9, a=2, c=0$, and a seed of 1, which produces a cycle of length 6. The second row is the same generator with a seed of 3, which produces a cycle of length 2. Using $a=4$ and $c=1$ (bottom row) gives a cycle length of 9 with any seed in [0, 8]. “

If  $c=0$, the generator is often called a multiplicative congruential method, or Lehmer RNG. If $c\neq0$ the generator is called a mixed congruential generator.

FAQs about linear congruential generator (LCG)

  1. What is meant by a linear congruential generator?
  2. How are random numbers generated?
  3. What are the conditions for linear congruential generators?
  4. What is meant by a multiplicative or mixed congruential generator?
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Read more about the pseudo-random process and Random number Generation

Read from Wikipedia about Linear Congruential Generator (LCG)