## RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

## Probability Distribution

Probability distribution is a function that gives the relative likelihood of occurrence of all possible outcomes of an experiment. There are two important functions that are used to describe a probability distribution. These are the probability density function or probability mass function and the cumulative distribution function.

In statistics, there can be two types of data, namely, discrete and continuous. Based on this, a probability distribution can be classified into a discrete probability distribution and a continuous probability distribution. In this article, we will learn more about probability distribution and the various aspects that are associated with it.

## What is Probability Distribution?

Probability distribution is a function that is used to give the probability of all the possible values that a random variable can take. A discrete probability distribution can be described by a probability distribution function and a probability mass function. Similarly, a probability distribution function and a probability density function are used to describe a continuous probability distribution. Binomial, Bernoulli, normal, and geometric distributions are examples of probability distributions.

## Probability Distribution Function

The probability distribution function is also known as the cumulative distribution function (CDF). If there is a random variable, X, and its value is evaluated at a point, x, then the probability distribution function gives the probability that X will take a value lesser than or equal to x. It can be written as F(x) = P (X ≤ x). Furthermore, if there is a semi-closed interval given by (a, b] then the probability distribution function is given by the formula P(a < X ≤ b) = F(b) – F(a). The probability distribution function of a random variable always lies between 0 and 1. It is a non-decreasing function.

## Probability Distribution Formulas

A probability distribution of a random variable can be described by a probability distribution function (CDF) and a probability mass function (for discrete random variables) or a probability density function (for continuous random variables). Depending upon the type of distribution a random variable follows there can be different formulas for a probability distribution.

### Probability Distribution of a Random Variable

A random variable can be described as a variable that can take on the possible values of an outcome of an experiment. There can be two types of random variables, namely, discrete and continuous random variables. Given below are the various formulas for the probability distribution of a random variable.

**Probability Distribution of a Discrete Random Variable**

A discrete random variable can be defined as a variable that can take a countable distinct value like 0, 1, 2, 3… The formulas for the probability distribution function and the probability mass function for a discrete random variable are given below:

- Probability Distribution Function: F(x) = P (X ≤ x)
- Probability Mass Function: p(x) = P(X = x)

**Probability Distribution of a Continuous Random Variable**

A continuous random variable can be defined as a variable that can take on infinitely many values. As the probability that a continuous random variable will take on an exact value is 0 hence, we cannot use the probability mass function (pmf) to describe such a distribution. We use the probability density function in place of the pmf. The formulas for the probability distribution of a continuous random variable are given below:

- Probability Distribution Function: F(x) = P (X ≤ x)
- Probability Density Function: f(x) = d/dx (F(x)) where F(x) =

### Probability Distribution of a Geometric Distribution

A geometric distribution is a type of discrete probability distribution where the random variable, X, represents the number of Bernoulli trials required till the first success is obtained. The outcome of each trial can either be a success (p) or a failure (1 – p). Given below are the formulas for the probability distribution of a geometric distribution.

- Probability Distribution Function: F(x) = P (X ≤ x) = 1 – (1 – p)
^{x} - Probability Mass Function: P(X = x) = (1 – p)
^{x – 1}p

### Probability Distribution of a Binomial Distribution

A binomial distribution is another type of discrete probability distribution that gives the number of successes when a sequence of n independent experiments is conducted. The outcome of each experiment can be either a success or a failure. It is denoted as X∼Bin(n,p)

The formulas for the probability distribution of a binomial distribution are given below:

- Probability Distribution Function: F(x) = P (X ≤ x) =

## Probability Distribution Graph

A probability distribution graph helps to give a visual approach of the distribution that a given random variable follows. For continuous distributions, the area under a probability distribution curve must always be equal to one. This is analogous to discrete distributions where the sum of all probabilities must be equal to 1.

For a continuous random variable, X, the probability density function is used to obtain the probability distribution graph. Suppose X lies between a and b, then the probability distribution graph is given as follows:

If a discrete random variable follows a probability distribution such as a Bernoulli distribution, then the probability distribution graph is given as follows:

The outcome of a Bernoulli trial can either be 0 or 1. Thus, the random variable, X, can only take on the values 0 or 1 as depicted in the graph.

## Probability Distribution Function and Probability Density Function

Both the probability distribution function and the probability density function are used to describe a probability distribution. A probability distribution function is used to summarize the probability distribution of a random variable. Such a function is well-defined for both continuous and discrete probability distributions. A probability density function (pdf), on the other hand, can only be used for continuous distributions. It can be defined as the likelihood that a continuous random variable, X, will take on a value that lies between a given range of values. For discrete distributions, a probability mass function (pmf) is used which is analogous to the probability density function. This function gives the probability that a random variable will exactly take on a specific value. We use the pdf in place of the pmf for continuous distributions as the probability that a continuous random variable will take on an exact value is 0.

**Important Notes on Probability Distribution**

- A probability distribution is used to describe all the possible values of a random variable and their corresponding occurrence probabilities.
- There can be two types of probability distributions. These are the continuous probability distribution (e.g., Normal distribution) and the discrete probability distribution (e.g., Bernoulli distribution).
- A probability distribution function and a probability density function (pdf) can be used to describe the characteristics of a continuous distribution.
- A discrete distribution can be defined by a probability mass function (pmf) and probability distribution function.

## Examples on Probability Distribution

**Example 2:** In a game of darts suppose you have a 25% chance that you will hit the bullseye. If you take a total of 15 shots then what is the probability that you will hit the bullseye 5 times?**Solution:** n = 15, p = 25 / 100 = 0.25, x = 5

We have to use the Binomial probability distribution given by P(X

## FAQs on Probability Distribution

### What is Meant by Probability Distribution?

Probability distribution is a statistical function that relates all the possible outcomes of a experiment with the corresponding probabilities.

### What is the Formula for a Probability Distribution?

There are two types of functions that are used to describe a probability distribution. These are the probability distribution function and the probability mass function (discrete random variable) or probability density function (continuous random variable). The formulas for these functions are given below:

- Probability Distribution Function: F(x) = P (X ≤ x)
- Probability Mass Function: p(x) = P(X = x)
- Probability Density Function: f(x) = d/dx (F(x)), where F(x) =

### What is the Probability Distribution of a Random Variable?

The probability distribution of a random variable describes how the probabilities of the outcomes of an experiment are distributed over the values of a random variable.

### What is Probability Distribution Function and Cumulative Distribution Function?

Probability distribution function and cumulative distribution function are the same. It is used to describe the distribution of both a continuous and discrete random variable.

### What is Probability Distribution Function and Probability Density Function?

Probability density function is only applicable to continuous random variables. It gives the probability that the value of a random variable will fall between a specified interval. The probability distribution function gives the probability that the value of a random variable will be less than or equal to a given outcome.

### What is Probability Distribution Used for?

Probability distribution is heavily utilized in determining confidence intervals and calculating critical regions for hypothesis testing (e.g., p-value). This is used in several industries that are driven towards data science.

## Formulas for Probability Distribution

The formulas for two types of the probability distribution are:

### Normal Probability Distribution Formula

It is also known as **Gaussian distribution** and it refers to the equation or graph which are bell-shaped.

The formula for normal probability distribution is as stated:

Where,

- μ = Mean
- σ = Standard Distribution.
*x*= Normal random variable.

**Note: **If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is known to be normal distribution.

### Binomial Probability Distribution Formula

It is defined as the probability that occurred when the event consists of “**n”** repeated trials and the outcome of each trial may or may not occur.

The formula for binomial probability is as stated below:

### Solved Probability Distribution Example Questions

**Question 1: **Calculate the probability of getting 8 tails, if a coin is tossed 10 times.

**Solution:**

Given,

Number of trails(n) = 10

Number of success(r) = 8(getting 8 tails)

Probability of single trail(p) =

**Question 2: Find the probability of normal distribution with population mean 2, standard deviation 3 of random variable 5.**

**Solution:**

Given,

x = 5

Mean = μ = 2

Standard deviation = σ = 3

Normal probability distribution:

**Question 3: The probability of a man hits the target is ¼. If he fires 9 times, then find the probability that he hits the target exactly 4 times.**

**Solution:**

Number of fires = n = 9

Number of success hits = r = 4

Probability of hitting the target = p = ¼

Probability of not hitting the target = q = 1 – p = 1 – (¼) = ¾

Finding ^{n}C_{r }:^{9}C_{4} = 9!/[4! (9-4)!] = 9!/(4! 5!) = (9 × 8 × 7 × 6 × 5!)/(4 × 3 × 2 × 1 × 5!) = 126

Probability of the person hits the target exactly 4 times

= ^{9}C_{4} (¼)^{4}(¾)^{(9-4)}

= 126 × (1/256) × (243/1024)

= 0.1168

## Probability Distribution of Random Variables

A random variable has a probability distribution, which defines the probability of its unknown values. Random variables can be discrete (not constant) or continuous or both. That means it takes any of a designated finite or countable list of values, provided with a probability mass function feature of the random variable’s probability distribution or can take any numerical value in an interval or set of intervals. Through a probability density function that is representative of the random variable’s probability distribution or it can be a combination of both discrete and continuous.

Two random variables with equal probability distribution can yet vary with respect to their relationships with other random variables or whether they are independent of these. The recognition of a random variable, which means, the outcomes of randomly choosing values as per the variable’s probability distribution function, are called **random variates.**

### Probability Distribution Formulas

## Types of Probability Distribution

There are two types of probability distribution which are used for different purposes and various types of the data generation process.

- Normal or Cumulative Probability Distribution
- Binomial or Discrete Probability Distribution

Let us discuss now both the types along with their definition, formula and examples.

## Cumulative Probability Distribution

The cumulative probability distribution is also known as a continuous probability distribution. In this distribution, the set of possible outcomes can take on values in a continuous range.

For example, a set of real numbers, is a continuous or normal distribution, as it gives all the possible outcomes of real numbers. Similarly, a set of complex numbers, a set of prime numbers, a set of whole numbers etc. are examples of Normal Probability distribution. Also, in real-life scenarios, the temperature of the day is an example of continuous probability. Based on these outcomes we can create a distribution table. A probability density function describes it. The formula for the normal distribution is;

Where,

- μ = Mean Value
- σ = Standard Distribution of probability.
- If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is known to be normal distribution.
- x = Normal random variable

### Normal Distribution Examples

Since the normal distribution statistics estimates many natural events so well, it has evolved into a standard of recommendation for many probability queries. Some of the examples are:

- Height of the Population of the world
- Rolling a dice (once or multiple times)
- To judge the Intelligent Quotient Level of children in this competitive world
- Tossing a coin
- Income distribution in countries economy among poor and rich
- The sizes of females shoes
- Weight of newly born babies range
- Average report of Students based on their performance

## Discrete Probability Distribution

A distribution is called a discrete probability distribution, where the set of outcomes are discrete in nature.

For example, if a dice is rolled, then all the possible outcomes are discrete and give a mass of outcomes. It is also known as the probability mass function.

So, the outcomes of binomial distribution consist of n repeated trials and the outcome may or may not occur. The formula for the binomial distribution is;

Where,

- n = Total number of events
- r = Total number of successful events.
- p = Success on a single trial probability.
^{n}C_{r}= [n!/r!(n−r)]!- 1 – p = Failure Probability

### Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In the real-life, the concept is used for:

- To find the number of used and unused materials while manufacturing a product.
- To take a survey of positive and negative feedback from the people for anything.
- To check if a particular channel is watched by how many viewers by calculating the survey of YES/NO.
- The number of men and women working in a company.
- To count the votes for a candidate in an election and many more.

## What is Negative Binomial Distribution?

In probability theory and statistics, if in a discrete probability distribution, the number of successes in a series of independent and identically disseminated Bernoulli trials before a particularised number of failures happens, then it is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e.r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.

## What is Poisson Probability Distribution?

The Poisson probability distribution is a discrete probability distribution that represents the probability of a given number of events happening in a fixed time or space if these cases occur with a known steady rate and individually of the time since the last event. It was titled after French mathematician Siméon Denis Poisson. The Poisson distribution can also be practised for the number of events happening in other particularised intervals such as distance, area or volume. Some of the real-life examples are:

- A number of patients arriving at a clinic between 10 to 11 AM.
- The number of emails received by a manager between office hours.
- The number of apples sold by a shopkeeper in the time period of 12 pm to 4 pm daily.

## Probability Distribution Function

A function which is used to define the distribution of a probability is called a Probability distribution function. Depending upon the types, we can define these functions. Also, these functions are used in terms of probability density functions for any given random variable.

In the case of **Normal distribution**, the function of a real-valued random variable X is the function given by;

**F _{X}(x) = P(X ≤ x)**

Where P shows the probability that the random variable X occurs on less than or equal to the value of x.

For a closed interval, (a→b), the cumulative probability function can be defined as;

**P(a<X ≤ b) = F _{X}(b) – F_{X}(a)**

If we express, the cumulative probability function as integral of its probability density function f_{X }, then,

In the case of a random variable X=b, we can define cumulative probability function as;

**In the case of Binomial distribution**, as we know it is defined as the probability of mass or discrete random variable gives exactly some value. This distribution is also called probability mass distribution and the function associated with it is called a probability mass function.

Probability mass function is basically defined for scalar or multivariate random variables whose domain is variant or discrete. Let us discuss its formula:

Suppose a random variable X and sample space S is defined as;

**X : S → A**

And A ∈ R, where R is a discrete random variable.

Then the probability mass function f_{X} : A → [0,1] for X can be defined as;

**f _{X}(x) = P_{r }(X=x) = P ({s ∈ S : X(s) = x})**

## Probability Distribution Table

The table could be created based on the random variable and possible outcomes. Say, a random variable X is a real-valued function whose domain is the sample space of a random experiment. The probability distribution P(X) of a random variable X is the system of numbers.

X | X_{1} | X_{2} | X_{3} | ………….. | X_{n} |

P(X) | P_{1} | P_{2} | P_{3} | …………… | P_{n} |

where Pi > 0, i=1 to n and P1+P2+P3+ …….. +Pn =1

## What is the Prior Probability?

In Bayesian statistical conclusion, a prior probability distribution, also known as the prior, of an unpredictable quantity is the probability distribution, expressing one’s faiths about this quantity before any proof is taken into the record. For instance, the prior probability distribution represents the relative proportions of voters who will vote for some politician in a forthcoming election. The hidden quantity may be a parameter of the design or a possible variable rather than a perceptible variable.

## What is Posterior Probability?

The posterior probability is the likelihood an event will occur after all data or background information has been brought into account. It is nearly associated with a prior probability, where an event will occur before you take any new data or evidence into consideration. It is an adjustment of prior probability. We can calculate it by using the below formula:

**Posterior Probability = Prior Probability + New Evidence **

It is commonly used in Bayesian hypothesis testing. For instance, old data propose that around 60% of students who begin college will graduate within 4 years. This is the prior probability. Still, if we think the figure is much lower, so we start collecting new data. The data collected implies that the true figure is closer to 50%, which is the posterior probability.

## Solved Examples

**Example 1**:

A coin is tossed twice. X is the random variable of the number of heads obtained. What is the probability distribution of x?

**Solution:**

First write, the value of X= 0, 1 and 2, as the possibility are there that

No head comes

One head and one tail comes

And head comes in both the coins

Now the probability distribution could be written as;

P(X=0) = P(Tail+Tail) = ½ * ½ = ¼

P(X=1) = P(Head+Tail) or P(Tail+Head) = ½ * ½ + ½ *½ = ½

P(X=2) = P(Head+Head) = ½ * ½ = ¼

We can put these values in tabular form;

X | 0 | 1 | 2 |

P(X) | 1/4 | 1/2 | 1/4 |

**Example 2: **

The weight of a pot of water chosen is a continuous random variable. The following table gives the weight in kg of 100 containers recently filled by the water purifier. It records the observed values of the continuous random variable and their corresponding frequencies. Find the probability or chances for each weight category.

Weight W | Number of Containers |

0.900−0.925 | 1 |

0.925−0.950 | 7 |

0.950−0.975 | 25 |

0.975−1.000 | 32 |

1.000−1.025 | 30 |

1.025−1.050 | 5 |

Total | 100 |

**Solution:**

We first divide the number of containers in each weight category by 100 to give the probabilities.

Weight W | Number of Containers | Probability |

0.900−0.925 | 1 | 0.01 |

0.925−0.950 | 7 | 0.07 |

0.950−0.975 | 25 | 0.25 |

0.975−1.000 | 32 | 0.32 |

1.000−1.025 | 30 | 0.30 |

1.025−1.050 | 5 | 0.05 |

Total | 100 | 1.00 |

## Frequently Asked Questions on Probability Distribution

### What is a probability distribution in statistics?

The probability distribution gives the possibility of each outcome of a random experiment. It provides the probabilities of different possible occurrences.

### What is an example of the probability distribution?

If two coins are tossed, then the probability of getting 0 heads is ¼, 1 head will be ½ and both heads will be ¼. So, the probability P(x) for a random experiment or discrete random variable x, is distributed as:

P(0) = ¼

P(1) = ½

P(2) = 1/4

P(x) = ¼ + ½ +¼ = 1

### What is the probability distribution used for?

The probability distribution is one of the important concepts in statistics. It has huge applications in business, engineering, medicine and other major sectors. It is majorly used to make future predictions based on a sample for a random experiment. For example, in business, it is used to predict if there will be profit or loss to the company using any new strategy or by proving any hypothesis test in the medical field, etc.

### What is the importance of Probability distribution in Statistics?

In statistics, to estimate the probability of a certain event to occur or estimate the change in occurrence and random phenomena modelled based on the distribution. Statisticians take a sample of the population to estimate the probability of occurrence of an event.

### What are the conditions of the Probability distribution?

There are basically two major conditions:

The probabilities for random events must lie between 0 to 1.

The sum of all the probabilities of outcomes should be equal to 1.

**Solved Example**

Here are some Probability Distribution examples that will help you to understand the concept thoroughly:

- What is the probability of getting 7 heads, if a coin is tossed for 12 times?

**Solution:**

Number of trials (n) =12

Number Of success (r) – 7

Probability of single-trial (p)= ½ = 0.5

- 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.

**Solution:**

Total Number of fires (n) =9

Total Number of success hites=r=4

**Fun Facts**

- New Probability theory was pioneered by Gerolama Cardano, Pierre de Fermat, and Blaise Pascal in the 16th century.
- The gambler dispute which took place in 1654 gave rise to the formation of the Mathematical Theory of Probability by two famous French Mathematicians Pierre de Fermat And Blaise Pascal
- Girloma Cardano is known as the “Father of Probability”.

**Quiz Time**

- Which is not possible in a Probability in the following types?
- p(x) = 0.5
- p(x) = -0.5
- p(x) = 1
- Σ x p(x) = 3

- In a Binomial Probability Distribution, if n is the number of trials and p is the Number of success, then the mean value is given by
^{n}p- n
- p
^{n}p_{(1-p)}

- Which of the following is not a feature of Normal Distribution?
- The mean value is always 0
- The area under the curve is equivalent to 1
- The mean, median, and mode are similar
- It is a symmetrical Distribution

- It is perfect to use Binomial Distribution for
- Large values of ‘n’
- Small values of ‘n’
- Fractional values of ‘n’
- Any values of ‘n’

## Để lại một phản hồi