39 Probability Distribution

V. Vimala

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1.   INTRODUCTION

 

In 17th century, the theory of probability was developed. The concept of probability was first developed from throwing a dice, games, tossing coins, drawing a card from a pack. In the year 1954, Antoine Gornband took an initiation to develop the probability distribution and made this area more interesting among the statisticians. In our routine life style the term probability or chance generally used. For instance, we say probably tomorrow the climate may be very hot in temperature, probably Ms.Vasanthi may come for party today, and probably you are right. These terms of possibility and probability express the same sense. But in statistics the probability has certain unique connotation unlike in Layman’s outlook.

 

2.   DEFINITION

 

Probability Distribution is distinct conditions of the principal sample space, with the set of all possible results of the random occurrence or happenings of the event which is generally observed. The sample space with the set of real numbers/ higher-dimensional vector space and non-numerical values are considered. For instance, the sample space of a coin flip will be either head or tail. The probability theory provides a means of getting an idea of the likelihood of occurrence of different events resulting from a random experiment in terms of quantitative measures ranging between zero and one. The probability is zero for an impossible event and one for an event which is certain to occur.

 

3.    PROBABILITY DISTRIBUTIONS

 

Probability Distributions are the record of all the values of the random variables. These variables are assumed as corresponding probabilities to create a probability distribution. Random variable doesn’t mean that all the values are different type or anything which is experimental study. These random variables are well explained with a set of results along with the well distinct probabilities for the happening of each outcomes of the explorative study or any research outcome. Here, the term random refers to the fact or statistics that the outcomes happen by chance or occurrence of events.

 

This can be well explained with a suitable example. For instance, the probability distribution which results in rolling of a dies in solo performance, it is discussed clearly in the below table – 1.

 

Table – 1: Single Fair Die

Mean, Variance, and Standard Deviation – To understand the probability distribution, it is very important to understand the mean, media, variance and standard deviation. From the following, it is very clearly specified that, how the population of the study can be distributed systematically.

 

Here, the definitions for population mean and variance are well defined for both the used ungrouped frequency distribution.

The above syntax can be explained by taking the population (N). This N – Population is divided by the population variance, the sample variance, which was the fair estimator for the population variance. This can be divided by n-1 for computation purpose.

 

After this process, you can use algebraic formulas or any equations to understand more effectively, this equation can be equivalent to:

To recollect, the probability are elongated term to the relative frequency distribution. So every frequency (f) and Population (N) [(f/N)] can be replaced by Probability (p) and Population (x) [p(x)].

 

This syntax can be simplified as follows:

To understand better, two formulas can be used for the last portion of variance in the mean square calculation.

 

The above example is worked out in the table – 2

 

Table – 2: Probability Distribution

The mean is xp(x) = (21/6) or 7/2 or 3.5,

Variance will be computed as follows –

=  x p(x) – x^2 p(x)

=  91/6 – (7/2)^2

=  15.17 – 12.25

=  2.92

The standard deviation is the square root of the variance = 1.7078

 

So, we can say that Probability Distribution (Pd) is a mathematical, arithmetical, numerical, statistical, geometrical function which can be stated in very simple way to make it very easy to provide the varies probabilities of occurrence of different possible happenings of the events in any experiment.

In more methodological terms, the probability distribution is a narrative of random happenings of the events which is represented in a pictorial way to understand the probabilities of occurrence of the events. The probability distribution is narrated in a bell shaped curve, with unimodal peaks at a single value. It is a Symmetrical where one side is mirror of the other which is mentioned with mean, median and mode in a bell shaped curve. (Mean=Md=Mo). The Pd is symptotic on both the left and right side of the normal bell shaped curve is asymptotic to x-axis, width is determined by quantum of amount with the variation of random variable. This is not used in all the business and it also replaced by Standardized Normal Distribution (SND) in general form.

 

This is a continuous distribution. It can be derived from the binomial distribution as a limiting case where n The no. of trials is very large. x & P the probability of success is close to ½. The general equation is

  f(x) = 1 e

√2п

-(x-μ)2

σ 2

 

Where the variable x lies between -∞ < x< ∞, μ & σ are called the parameters of the distribution. F(x) is called pdf of the normal distribution N( μ ,σ 2 ).The graph of the normal distribution is called the normal curve. It is bell shaped and symmetric about its mean. The two tails of the curve extend to +∞ & -∞ the curve is unimodal. The total area under the curve is 1.

 

EXAMPLE –

(a) A coin is tossed 3 times. Find the probability of getting 2 heads and a tail in any given order.

 

3.1. Normal Random Variable/Normal Distribution (NRC/ND)

 

The normal random variables or distributions can be described as continuous random variables. It can be better understood by two predominant ways which is represented below -(a) Density plot – Shape : Bell

3.2.   Standard Normal Distribution (SND)

 

The Normal distribution with mean zero (mean=0) and standard deviation ( s /SD=1). It denotes N (0, 1). Normally we use Z to denote a standard normal random variable. It is very important to know that for what purpose we learn or understand the SND. To calculate the area under the normal curve either numerically or geometrically. Many statisticians have established tables to indicate the left tail area under the Standard Normal Curve (SNC) of any given number. Probability distributions are a elementary model in statistics. They are used both on a theoretical point and a practical point.

 

3.3.  Discrete Probability Distribution (DPD) –

 

It applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function.

 

3.4.  Continuous Probability Distribution (CPD) –

 

It applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g., real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution.

 

4.  Random variable

 

Random variable is a variable whose value is determined by the outcome of a random experiment. Random variable whose value is determined by the outcome of a random experiment is called a random variable. An example of this is the income of a randomly selected family. A random variable X is said to have the normal distribution with parameters µ and σ if its density function is given by: f(x) = 1 √ 2π σ exp (− 1 2 x − µ σ 2) (6) for −∞ < x < ∞. It can be shown that E(X) = µ and V (X) = σ 2. Thus, the normal distribution is characterized by a mean µ and a standard deviation σ.

 

4.1.  Discrete Random Variable

 

A discrete random variable is one whose set of assumed values is countable (arises from counting). Discrete random variable whose values are countable is called a discrete random variable. An example of this is the number of cars in a parking lot at any particular time.

 

4.2.  Continuous Random Variable

 

A continuous random variable is one whose set of assumed values is uncountable (arises from measurement.).Continuous random variable that can assume any value in one or more intervals is called a continuous random variable. An example of this is the time taken by a person to travel by car from New York City to Boston.

 

5. BINOMIAL DISTRIBUTION

 

In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

 

EXAMPLE –

An elementary example is this: roll a die ten times and count the number of 1s a outcome. Then this random number follows a binomial distribution with n = 10 and p =1/6.

 

For example, assume 5% of the population is green-eyed. You pick 500 people randomly. The number of green-eyed people you pick is a random variable X which follows a binomial distribution with n = 500 and p = 0.05.

 

6.    CONDITIONAL PROBABILITY

The conditional probability of two events A and B is P (A|B) =P( A   and P(B) B)

where P (A and B) means the probability of the outcomes that events A and B have in common.

 

(i)  EXAMPLE –

When a die is rolled once, find the probability of getting a 4 given that an even number occurred in an earlier throw.

 

Solution:

 

P (4 and an even number) = 1/6

.i.e. P (A and B) =1/6.

P (even number) =3/6 =1/2.

   (ii)     EXAMPLE –

 

A bag contains 3 orange, 3 yellow and 2 white marbles. Three marbles are selected without replacement. Find the probability of selecting two yellow and a white marble.

 

Solution:

 

P( 1st Y) =3/8, P( 2nd Y) = 2/7 and P( W)= 2/6

P(Y and Y and W) =P(Y) x P(Y) x P (W) = 3/8 x 2/7 x 2/6 = 1 / 28

 

(iii)    In a class, there are 8 girls and 6 boys. If three students are selected at random for debating, find the probability that all girls.

 

Solution:

P (G) =8/14 and P (B) =6/14. P (1st G) =8/14, P (2nd G) 7/13 and P (3rdG) = 6/12. P (three girls) 8/14 x 7/13 x 6/12= 2/13

 

(iv)In how many ways can 3 drama officials be selected from 8 members?

Solution:             8C3 = 56 ways.

 

(v)  A box has 12 bulbs, of which 3 are defective. If 4 bulbs are sold, find the probability that exactly one will be defective.

 

Solution:

 

P (defective bulb) = 3C1 and P (non-defective bulbs) = 9C3

3C1 x9C3 = 3!x 9!= 252

(3 -1)!1! (9 – 3)!3!

P (4 bulbs from 12) = 12C4 = 495.

P (1 defective bulb and 3 okay bulbs) = 295/495=0.509.

 

7.    POISSON DISTRIBUTION

 

Poisson distribution: is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. The Poisson distribution arises in many situations. It is safe to say that it is one of the three most important discrete probability distributions (the other two being the uniform and the binomial distributions). The Poisson distribution can be viewed as arising from the binomial distribution or from the exponential density.

 

If the parameters n & p of a binomial distribution are known then we can find the distribution. But when n is large and p is very small the application of binomial distribution is very difficult. Let x be any discrete random variable which can take values 0,1,2,3….. such that the probability distribution function of x

 

P(x)=e -λ λx x!

 

where λ is a positive constant, np = λ . This distribution is called the poisson distribution.

 

Example –

 

 Number of printing mistakes on each page of a book published by a good publisher

 

Number of telephone calls arriving at a telephone switch board per minute

 

The Poisson distribution is a common distribution used to model “count” data:

  • Number of telephone calls received per hour
  • Number of claims received per day by an insurance company
  • Number of accidents per month at an intersection

8.  OBJECTIVES OF PROBABILITY DISTRIBUTION The core objectives of tabulation are mentioned below:

 

(a)To compute and interpret the expected value, variance, and standard deviation for a discrete random variable and work with probabilities involving a binomial probability distribution.

 

(b)To work out the probabilities involving a Poisson probability distribution,

 

(c)Understand the concepts of a random variable and a probability distribution.

 

(d) To decide the binomial distribution problems to be approximated by the Poisson distribution , to use the Poisson distribution in analyzing statistical

 

(e)To use the hyper geometric distribution and know how to work such problems.

 

9. RULES OF PROBABILITY

 

(a)   ADDITION RULES I. Rule – 1:

 

When two events A and B are mutually exclusive, then P (A or B) =P (A) +P (B)

 

Example –

 

When a is tossed, find the probability of getting a 3 or 5.

 

Solution: P (3) =1/6 and P (5) =1/6.

 

Therefore P (3 or 5) = P (3) + P (5) = 1/6+1/6 =2/6=1/3.

 

II. Rule – 2:

If A and B are two events that are NOT mutually exclusive, then P (A or B) = P (A)

 

+   P (B) – P (A and B), where A and B means the number of outcomes that event A and B have in common.

 

Example –

 

When a card is drawn from a pack of 52 cards, find the probability that the card is a 10 or a heart.

 

Solution:

 

P (10) = 4/52 and P (heart) =13/52

 

P (10 that is Heart) = 1/52

 

P (A or B) = P (A) +P (B) – P (A and B) = 4/52 _ 13/52 – 1/52 = 16/52.

 

(b) MULTIPLICATION RULES

 

I. Rule – 1: For two independent events A and B, then P (A and B) = P (A) x P (B).

 

Example –

 

Determine the probability of obtaining a 5 on a die and a tail on a coin in one throw.

 

Solution: P (5) =1/6 and P (T) =1/2.

 

P (5 and T) = P (5) x P (T) = 1/6 x ½= 1/12.

 

II. Rule – 2:

 

When to events are dependent, the probability of both events occurring is P(A and B)=P(A) x P(B|A), where P(B|A) is the probability that event B occurs given that event A has already occurred.

 

Example –Find  the  probability  of  obtaining  two  Aces  from  a  pack  of  52  cards  without replacement.

 

Solution:

 

P( Ace) =2/52 and P( second Ace if NO replacement) = 3/51

 

Therefore P (Ace and Ace) = P(Ace) x P( Second Ace) = 4/52 x 3/51 = 1/221

 

10. MERITS OF PROBABILITY DISTRIBUTION

 

Merits of probability distribution and benefits of theoretical or probability distribution

  • Where the calculation of observed distribution is not possible.
  • Sometime observed base distribution’s calculation is not possible due to impossibility at that place we can calculate probability distribution.
  • Helpful in forecasting – Probability distribution isvery helpful for forecasting and on this basis we can estimate our future and make good plans for our business.
  • Helpful in Comparison – It can compare it with observed or real distribution and evaluate our efficiency of work.
  1. LIMITATIONS OF PROBABILITY DISTRIBUTION
  • Probability distribution cannot be correlated
  • Probability distribution cannot be viewed in chart form or in any statistical view
  • The data cannot be extract data from probability distribution or it cannot include in the reports
  • It is excluded from sensitivity analyses or charts.
  • Probability distribution doesn’t support the Latin Hypercube sampling
  1. CONCLUSION

To summaries, the probability has become one of the basic tools of statistics. Sometimes statistical analysis becomes paralyzed without the theorem of probability. Probability of a given event is defined as the expected frequency of occurrence of the event among events of a like sort. According to basic economic theory, people wish to maximize their expected utility. In order to do so they should integrate the likelihood (i.e. probability) and the possible outcomes (good or bad). This means that people maximize their utility based on their perceived importance of probabilities and outcomes.

 

you can view video on Probability Distribution

Web links

  • https://www.merriam-webster.com/dictionary/probability
  • http://whatis.techtarget.com/definition/probability
  • http://www.itl.nist.gov/div898/handbook/eda/section3/eda361.htm
  • http://study.com/academy/lesson/probability-distribution-definition-formula-example.html
  • https://www.intmath.com/counting-probability/11-probability-distributions-concepts.php
  • https://onlinecourses.science.psu.edu/stat200/node/34
  • https://www.thoughtco.com/probability-distribution-3126569
  • https://people.richland.edu/james/lecture/m170/ch06-prb.html
  • https://www.utdallas.edu/~scniu/OPRE-6301/documents/Important_Probability_Distributions.pdf
  • http://www.ce.memphis.edu/7012/pdf%20files/Discrete_notes.pdf
  • https://cran.r-project.org/web/packages/IPSUR/vignettes/IPSUR.pdf
  • http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
  • https://www.intmath.com/counting-probability/11-probability-distributions-concepts.php
  • http://study.com/academy/lesson/probability-distribution-definition-formula-example.html http://www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm
  • https://www.slideshare.net/bijayabnanda/ls-bs-9probability-concept-and-probability-distribution
  • https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability_dis tributions