Theoretical Distribution

In the real world, experiments with single outcomes like heads or tails are rare; generally, experiments
are collection of set of events and carried by n number of times, the of outcome which, represents a
form of theoretical distribution. A random exponent assumed in theoretical distribution, given by a
function of the random variable, called probability function.
For example, if we toss a fair coin, the probability of getting a head is 12. If we toss it for 50 times,
the probability of getting a head is 25. We call this as the theoretical or expected frequency of the
heads. Actually, by tossing a coin, we may get 25, 30 or 35 heads, which we call as the observed
frequency.
Thus, the observed frequency and the expected frequency may equal or may differ from each other
due to fluctuation in the experiment.

Types of Theoretical Distribution

1. Binomial Distribution

2. Poisson distribution

3. Normal distribution or Expected Frequency distribution

Binomial Distribution:
The prefix ‘Bi’ means two or twice. A binomial distribution is the probability of a trail with two and only
two outcomes. It is a type of distribution that has two different outcomes namely, ‘success’ and
‘failure’. In addition, it is applicable to discrete random variables only. In binomial probability
distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question
is asked for yes-no, then the Boolean-valued outcome is represented either with
success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single
success/failure test called a Bernoulli trial or Bernoulli experiment, and a series of outcomes called
a Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli
distribution. The binomial distribution is the base for the famous binomial test of statistical
importance.

The three different criteria of binomial distributions are:

1. The number of the trial are fixed.

2. Every trial is independent that is, not effected by next trial.

3. The probability always stays the same and equal.

Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by;
 P(x:n,p) = nCx px (1-p)n-x

Or
P(x:n,p) = nCx px (q)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials, where nCx =
n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

Binomial Distribution Mean and Variance

For a binomial distribution, the mean, variance and standard deviation for the given number of
success represented using the formulas
Mean, μ = np
Variance, σ2 = npq
Standard Deviation σ= √(npq)
Where p is the probability of success
q is the probability of failure, where q = 1-p

Properties of Binomial Distribution

The properties of the binomial distribution are:

 There are two possible outcomes: true or false, success or failure, yes or no.
 There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
 The probability of success or failure varies for each trial.
 Only the number of success calculated, out of ‘n’ independent trials.
 Every trial is an independent trial, which means the outcome of one trial does not affect
the outcome of another trial.

Applications of binominal distributions

 Finding the quantity of raw and used materials while making a product.
  Taking a survey of positive and negative reviews from the public for any specific product
or place.
 By using the YES/ NO survey, we can check whether the number of persons views the
particular channel.
 To find the number of male and female employees in an organization.
 The number of votes collected by a candidate in an election is counted based on 0 or 1
probability.

Binomial Distribution Examples and Solutions

Example 1: If a coin is tossed 5 times, find the probability of ; Exactly 2 heads
Solution:
The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:
Number of trials: n=5
Probability of head: p= 1/2 and hence the probability of tail, q =1/2
For exactly two heads:
x=2
P(x=2) = 5C2 p2 q5-2 = 5! / 2! 3! × (½)2× (½)3
P(x=2) = 5/16
Example 2:
A fair coin is tossed 10 times, what are the probability of getting exactly 6 heads and at least six
heads.
Solution:
Let x denote the number of heads in an experiment.
Here, the number of times the coin tossed is 10. Hence, n=10.
The probability of getting head, p ½
The probability of getting a tail, q = 1-p = 1-(½) = ½.
The binomial distribution is given by the formula:
P(X= x) = nCxpxqn-x, where = 0, 1, 2, 3, …
Therefore, P(X = x) = 10Cx(½)x(½)10-x
(i) The probability of getting exactly 6 heads is:
P(X=6) = 10C6(½)6(½)10-6
P(X= 6) = 10C6(½)10
P(X = 6) = 105/512.
Hence, the probability of getting exactly 6 heads is 105/512.
Poisson Distribution Definition
The Poisson distribution is a discrete probability function that means the variable can only take
specific values in a given list of numbers, probably infinite. A Poisson distribution measures how
many times an event is likely to occur within “x” period of time. In other words, we can define it as
the probability distribution that results from the Poisson experiment. A Poisson experiment is a
statistical experiment that classifies the experiment into two categories, such as success or
failure. Poisson distribution is a limiting process of the binomial distribution. The Poisson
Distribution is a theoretical discrete probability distribution that is very useful in situations where the
events occur in a continuous manner. Poisson Distribution is utilized to determine the probability of
exactly x0 number of successes taking place in unit time. Let us now discuss the Poisson Model.

At first, we divide the time into n number of small intervals, such that n → ∞ and p denote the
probability of success, as we have already divided the time into infinitely small intervals so p → 0. So
the result must be that in that condition is n x p = λ (a finite constant).

A Poisson random variable “x” defines the number of successes in the experiment. This
distribution occurs when there are events that do not occur as the outcomes of a definite
number of outcomes. Poisson distribution is used under certain conditions. They are:

 The number of trials “n” tends to infinity

 Probability of success “p” tends to zero
 np = 1 is finite

Poisson Distribution Formula

The formula for the Poisson distribution function is given by:
f(x) =(e– λ λx)/x!
Where,
e is the base of the logarithm
x is a Poisson random variable
λ is an average rate of value

Poisson Distribution Mean and Variance

Assume that, we conduct a Poisson experiment, in which the average number of successes
within a given range is taken as λ. In Poisson distribution, the mean of the distribution is
represented by λ and e is constant, which is approximately equal to 2.71828. Then, the Poisson
probability is:
P(x, λ ) =(e– λ λx)/x!
In Poisson distribution, the mean is represented as E(X) = λ.
 For a Poisson Distribution, the mean and the variance are equal. It means that E(X) = V(X)
Where,
V(X) is the variance.

Poisson Distribution Expected Value

A random variable is said to have a Poisson distribution with the parameter λ, where “λ” is
considered as an expected value of the Poisson distribution.
The expected value of the Poisson distribution is given as follows:
E(x) = μ = d(eλ(t-1))/dt, at t=1.
E(x) = λ
Therefore, the expected value (mean) and the variance of the Poisson distribution is equal to λ.

Properties of Poisson distribution:

 The event or success is something that can be counted in whole numbers.

 The probability of having success in a time interval is independent of any of its

previous occurrence.

 The average frequency of successes in a unit time interval is known.

 The probability of more than one success in unit time is very low.

Applications
 Number of Network Failures per Week.
 Number of Bankruptcies Filed per Month.
 Number of Website Visitors per Hour.
 Number of Arrivals at a Restaurant.
 Number of Calls per Hour at a Call Center.
 Number of Books Sold per Week.
 Average Number of Storms in a City.

Poisson Distribution Examples

An example to find the probability using the Poisson distribution is given below:
Example 1:
A random variable X has a Poisson distribution with parameter λ such that P (X = 1) = (0.2) P (X
= 2). Find P (X = 0).
Solution:
For the Poisson distribution, the probability function is defined as:
P (X =x) = (e– λ λx)/x!, where λ is a parameter.
Given that, P (x = 1) = (0.2) P (X = 2)
(e– λ λ1)/1! = (0.2)(e– λ λ2)/2!
⇒λ = λ2/ 10
⇒λ = 10
Now, substitute λ = 10, in the formula, we get:
P (X =0 ) = (e– λ λ0)/0!
P (X =0) = e-10 = 0.0000454
Thus, P (X= 0) = 0.0000454

Normal Distribution:
The Normal Distribution defines a probability density function f(x) for the continuous random
variable X considered in the system. The random variables, which follow the normal distribution, are
ones whose values can assume any known value in a given range.

We can hence extend the range to – ∞ to + ∞ . Continuous Variables are such random variables
and thus, the Normal Distribution gives you the probability of your value being in a particular range
for a given trial. The normal distribution is very important in the statistical analysis due to the central
limit theorem.

The theorem states that any distribution become normally distributed when the number of variables
is sufficiently large. For instance, the binomial distribution tends to change into the normal distribution
with mean and variance.

In probability theory and statistics, the Normal Distribution, also called the Gaussian
Distribution, is the most significant continuous probability distribution. Sometimes it is also called
a bell curve. A large number of random variables are either nearly or exactly represented by the
normal distribution, in every physical science and economics. Furthermore, it can
be used to approximate other probability distributions, therefore supporting the usage of the word
‘normal ‘as in about the one, mostly used.

Normal Distribution Definition

The Normal Distribution is defined by the probability density function for a continuous random
variable in a system. Let us say, f(x) is the probability density function and X is the random
variable. Hence, it defines a function which is integrated between the range or interval (x to x +
dx), giving the probability of random variable X, by considering the values between x and x+dx.
f(x) ≥ 0 ∀ x ϵ (−∞,+∞)
And -∞∫+∞ f(x) = 1

Normal Distribution Formula

The probability density function of normal or gaussian distribution is given by;

Where,

 x is the variable
 μ is the mean
 σ is the standard deviation

Normal Distribution Curve

The random variables following the normal distribution are those whose values can find any
unknown value in a given range. For example, finding the height of the students in the school.
Here, the distribution can consider any value, but it will be bounded in the range say, 0 to 6ft.
This limitation is forced physically in our query.
Whereas, the normal distribution doesn’t even bother about the range. The range can also
extend to –∞ to + ∞ and still we can find a smooth curve. These random variables are called
Continuous Variables, and the Normal Distribution then provides here probability of the value
lying in a particular range for a given experiment. Also, use the normal distribution calculator to
find the probability density function by just providing the mean and standard deviation value.

Normal Distribution Standard Deviation

Generally, the normal distribution has any positive standard deviation. We know that the mean
helps to determine the line of symmetry of a graph, whereas the standard deviation helps to
know how far the data are spread out. If the standard deviation is smaller, the data are
somewhat close to each other and the graph becomes narrower. If the standard deviation is
larger, the data are dispersed more, and the graph becomes wider. The standard deviations are
used to subdivide the area under the normal curve. Each subdivided section defines the
percentage of data, which falls into the specific region of a graph.
Using 1 standard deviation, the Empirical Rule states that,

 Approximately 68% of the data falls within one standard deviation of the mean. (i.e.,
Between Mean- one Standard Deviation and Mean + one standard deviation)
 Approximately 95% of the data falls within two standard deviations of the mean. (i.e.,
Between Mean- two Standard Deviation and Mean + two standard deviations)
  Approximately 99.7% of the data fall within three standard deviations of the mean. (i.e.,
Between Mean- three Standard Deviation and Mean + three standard deviations)

Thus, the empirical rule is called the 68 – 95 – 99.7 rule.

Normal Distribution Properties

Some of the important properties of the normal distribution are listed below:

 In a normal distribution, the mean, mean and mode are equal.(i.e., Mean = Median=
Mode).
 The total area under the curve should be equal to 1.
 The normally distributed curve should be symmetric at the centre.
 There should be exactly half of the values are to the right of the centre and exactly half
of the values are to the left of the centre.
 The normal distribution is, defined by the mean and standard deviation.
 The mean and median are the same and lie in the middle of the distribution

 The normal distribution curve must have only one peak. (i.e., Unimodal)
 The curve approaches the x-axis, but it never touches, and it extends farther away from
the mean.
 About 68 percent of its values lie within one standard deviation of the mean.
 About 95 percent of its values lie within two standard deviations of the mean.

 Almost all of its values lie within three standard deviations of the mean.
Applications
The normal distributions are closely associated with many things such as:

 Marks scored on the test

 Heights of different persons
 Size of objects produced by the machines
 Blood pressure

Normal Distribution Problems and Solutions

Question 1: Calculate the probability density function of normal distribution using the
following data. x = 3, μ = 4 and σ = 2.
Solution: Given, variable, x = 3
Mean = 4 and
Standard deviation = 2
By the formula of the probability density of normal distribution, we can write;

Hence, f(3,4,2) = 1.106.

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