In Probability distribution, a probability is assigned to a measurable subset of the possible outcomes of any of the random experiment or the survey. The sample space of the experiments are of two types. The Discrete random variables and the continuous random variable.

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We know that random variables can be divided into two: Discrete and Continuous.

A discrete random variable can take only a finite number of different values like 0,1,2,3,4,..and so on, whereas a continuous random variable is a variable that can take an infinite number of possible values. Continuous random variables are usually measurements, like height, weight etc.

Probability distribution for a discrete random variable:

A probability distribution function (p.d.f) for any discrete random variable is the values of all the probable numerical results for that random variable provided that they are mutually exclusive, such that each probability is associated with each of the outcome.

The p.d.f can be defined as follows:

**1.** The probability distribution 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, $p_{i}> 0$, and $\sum_{i=1}^{n}p_{i}=1$ where i = 1, 2,..., n.

**2.** The probabilities $p_{i}$, where P(X = x_{i}) = pi, must satisfy the following:

1: $0< p_{i}< 1$ for each i

2: $p_{1}, p_{2}, p_{3},...p_{n}$= 1.

To summarize, a Discrete Probability Distributions have 3 major properties:

1) $\sum P(X)=1$

2) $\sum P(X)\geq 0$

3) When we substitute the random variable into the function, we find out the probability that the particular value will occur.

Some of the important major probability distributions are: Binomial distribution, Hyper geometric distribution and Poisson distribution.

**Probability distribution for a continuous random variable:**

Continuous Probability distributions are also known as Probability density function. The probability is interpreted as "area under the curve."

It has the following properties:

1) The random variable takes on an infinite number of values within a given interval

2) The probability that X equals to any particular value is 0, and hence, here intervals are considered. The probability is calculated as, the area under the curve.

3) The area under the whole curve = 1.

The formula for the probability in which the random variable x takes the value in the interval which is $a_{1}\leq x\leq a_{2}$ is given as $\int_{a_{1}}^{a_{2}}p(x)dx$

Some of the continuous probability distributions are: Normal distribution, Standard Normal distribution, Student's t distribution, Chi-square distribution and F distribution.

**Solved Examples:**

**Example 1:** Find the Probability Distribution for the Toss of a Die:

**Solution:** The probability distribution for the toss of a die is as follows:

$X_{i}$ P($X_{i}$)

1 $\frac{1}{6}$

2 $\frac{1}{6}$

3 $\frac{1}{6}$

4 $\frac{1}{6}$

5 $\frac{1}{6}$

6 $\frac{1}{6}$

This example is an example of a uniform distribution.

**Example 2:** If literacy is Normally Distributed with a mean of 100 and a standard deviation of 10, what percentage of the population will have

(a) Literacy ranging from 90 to 110?

(b) Literacy ranging from 80 to 120?

**Solution:** To find these probabilities, we will have to compute the z-scores (as this is a case of standard normal distribution) for each of these as:

(a) Z = $\frac{90-100}{10}$ = -1 area = .3413

Z = $\frac{110-100}{10}$ = +1 area = .3413

0.6826

Hence, 68.26% of the population.

(b) Z = $\frac{80-100}{10}$ = -2 area = .4772

Z = $\frac{120-100}{10}$ = +2 area = .4772

.9544

Hence, 95.44% of the population.

A discrete random variable can take only a finite number of different values like 0,1,2,3,4,..and so on, whereas a continuous random variable is a variable that can take an infinite number of possible values. Continuous random variables are usually measurements, like height, weight etc.

Probability distribution for a discrete random variable:

A probability distribution function (p.d.f) for any discrete random variable is the values of all the probable numerical results for that random variable provided that they are mutually exclusive, such that each probability is associated with each of the outcome.

The p.d.f can be defined as follows:

X : $x_{1}, x_{2}, x_{3},...x_{n}$

P(X) : $p_{1}, p_{2}, p_{3},...p_{n}$

where, $p_{i}> 0$, and $\sum_{i=1}^{n}p_{i}=1$ where i = 1, 2,..., n.

1: $0< p_{i}< 1$ for each i

2: $p_{1}, p_{2}, p_{3},...p_{n}$= 1.

To summarize, a Discrete Probability Distributions have 3 major properties:

1) $\sum P(X)=1$

2) $\sum P(X)\geq 0$

3) When we substitute the random variable into the function, we find out the probability that the particular value will occur.

Some of the important major probability distributions are: Binomial distribution, Hyper geometric distribution and Poisson distribution.

Continuous Probability distributions are also known as Probability density function. The probability is interpreted as "area under the curve."

It has the following properties:

1) The random variable takes on an infinite number of values within a given interval

2) The probability that X equals to any particular value is 0, and hence, here intervals are considered. The probability is calculated as, the area under the curve.

3) The area under the whole curve = 1.

The formula for the probability in which the random variable x takes the value in the interval which is $a_{1}\leq x\leq a_{2}$ is given as $\int_{a_{1}}^{a_{2}}p(x)dx$

Some of the continuous probability distributions are: Normal distribution, Standard Normal distribution, Student's t distribution, Chi-square distribution and F distribution.

$X_{i}$ P($X_{i}$)

1 $\frac{1}{6}$

2 $\frac{1}{6}$

3 $\frac{1}{6}$

4 $\frac{1}{6}$

5 $\frac{1}{6}$

6 $\frac{1}{6}$

This example is an example of a uniform distribution.

(a) Literacy ranging from 90 to 110?

(b) Literacy ranging from 80 to 120?

(a) Z = $\frac{90-100}{10}$ = -1 area = .3413

Z = $\frac{110-100}{10}$ = +1 area = .3413

0.6826

Hence, 68.26% of the population.

(b) Z = $\frac{80-100}{10}$ = -2 area = .4772

Z = $\frac{120-100}{10}$ = +2 area = .4772

.9544

Hence, 95.44% of the population.