Mostly, statisticians make use of capital letters to denote a Probability Distribution of random variables and small-case letters to represent any of its values.
X denotes the Probability Distribution of random variable X
P(X) denotes the Probability of X.
p(X = x) denotes the Probability that random variable x is equivalent to any particular value, represented by X. For example: P (X = 1) states the Probability Distribution of the random variable X is equivalent to 1.
There are two types of Probability Distribution which are used for distinct purposes and various types of data generation processes.
Normal Probability Distribution
In this Distribution, the set of all possible outcomes can take their values on a continuous range. It is also known as Continuous or cumulative Probability Distribution.
For example- Set of real Numbers, set of prime numbers, are the Normal Distribution examples as they provide all possible outcomes of real Numbers and Prime Numbers. Real-life scenarios such as the temperature of a day is an example of Continuous Distribution.
As the Normal Distribution Statistics predict some natural events clearly, it has developed a standard of recommendation for many Probability issues. Some examples are:
Binomial / Discrete Probability Distribution
The Binomial Distribution is also termed as a Discrete Probability Function where the set of outcomes is Discrete in nature. For example: if a dice is rolled, then all its possible outcomes will be Discrete in nature and it gives the mass of outcome. It is also considered a Probability mass Function
A negative Binomial Distribution is a term used when in a given Discrete Probability Distribution, before a particularized Number of failures occurs, the Number of the success in the series of the independent and identical Bernoulli trials happens. The Number of failures here is denoted by the letter ‘r’. For example, while throwing a dice, we determine the occurrence of the Number 1 as a failure and all the mom-1’s as a success. Now, throwing the dice Continuously until the Number 1 occurs three times, indicating three failures, in this case, the Probability Distribution of the non-1 Numbers that have arrived would be referred to as the Negative Binomial Distribution.
This Discrete Probability Distribution presents the Probability of a given number of events that occur in time and space, at a steady rate. It had gained its name from the French Mathematician Simeon Denis Poisson. This kind of Distribution also finds its relevance in other events occurring at particular intervals, for example, distance, area, and volume. Some examples of these are;
Prior Probability, also known as prior, of a quantity that is unpredictable, refers to the Probability Distribution which expresses one’s faith in the given quantity before any given proof is taken into records. For example, the prior Probability Distribution points at the relative proportions of voters that might vote for a given politician at the election. The hidden quantity can point at the possible variable rather than at a perceptible variable.
Here are some of the Probability Distribution formulas based on their types.
The Formula for the Normal Distribution
Here,
μ = Mean Value
σ =Standard Deviation
x = Normal random variable
If mean μ = 0, and standard deviation = 1, then this Distribution is termed as Normal Distribution.
The Formula for the Binomial Distribution
Here,
n=Total Number of events
r= Total Number of successful events
p = successful on a single trial Probability,
1-p = Failure Probability
nCr = n!r!(n−r)!
The Functions which are used to define the Distribution of Probability are termed as a Probability Distribution Function. These Functions can be defined on the basis of their types. These Probability Distribution Functions are also used in respect of Probability Density Functions for any of the given random variables.
In Normal Distribution, the Function of a real-valued random variable X is the Function derived by:
Fx(x) =P(X ≤ x)
Where P indicates the Probability that the random variable X occurs on less than or equal to the value of X.
For the closed interval (a →b) the cumulative Probability Function can be identified as:
P( a< X ≤ b) = Fx (b) -Fx(a)
If the cumulative Probability Function is expressed as integral of the Probability density Function fx, then,
In terms of a random variable X= b, cumulative Probability Function can be defined as:
As we know, the Binomial Distribution is determined as the Probability of mass or Discrete random variable which yields exactly some values. This Distribution is also termed Probability mass Distribution and the Function linked with it is known as Probability mass Function.
For example,
A random variable X and sample space S are termed as
X:S → A
And A ∈ R, where R is termed as a Discrete random variable
Then, Probability mass Function fx : A -
0,1
0,1 or X can be termed as:
Fx (x) = Pr(X = x) = P ({s ∈ S: X(s) = x})
The Probability Distribution table is designed in terms of a random variable and possible outcomes. For instance- random variable X is a real-valued function whose domain is considered as the sample space of a random experiment. The Probability Distribution of P(X) of a random variable X is the arrangement of Numbers.
Where Pi > 0 , i = 1 to n and P1 + P2 + P3 ….. Pn = 1
Where Pi > 0 , i = 1 to n and P1 + P2 + P3 ….. Pn = 1
Example 1: What is the probability of getting 7 heads, if a coin is tossed for 12 times?
Number of trials (n) =12
Number Of success (r) - 7
Probability of single-trial (p)= ½ = 0.5nCr
= n!/r! X (n-r)!
=12! /7! (12-7)!
= 12! / 7! 5!
= 95040120
= 792
pr = 0.5 = 0.0078125To find(1−p)(n−r), calculate (1-p) and (n-r)
(1-p) =1-0.5 = 0.5
n-r = 12-7= 5
(1−p)(n−r) = (0.5)(7) = 0.03125Now calculate
P(X = r)(nCr.p)r.(1−p)n−r
= 792 x 0.0078125 x 0.03125
= 0.193359375
Hence, the Probability of getting 7 head is 0.19
Example 2: The Probability of a man hitting the target is ¼. If he fires 9 times, then find the Probability that he hits the target exactly 4 times.
Total Number of fires (n) = 9
Total Number of success hites = r = 4
Probability of hitting the targets
Probability of not hitting the targets
Calculating nCr
Probability of the person hits the target exactly 4 times
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