what is difference between discrete Probability distribution and

The difference between discrete probability distribution and continuous probability distribution lies in the nature of the variables they represent. Let's explore each one and provide an example for better understanding.
**Discrete Probability Distribution:**
A discrete probability distribution is associated with discrete random variables. Discrete random variables are those that can take on a finite or countably infinite set of values. The probability distribution assigns probabilities to each possible value of the random variable.
To illustrate this, let's consider an example. Suppose we have a fair six-sided die, and we want to know the probability of rolling each possible number. The random variable in this case would be the outcome of the roll, which can take on the values 1, 2, 3, 4, 5, or 6. Since each outcome has an equal chance of occurring, the probability distribution would be as follows:
$P(X=1)=\frac{1}{6},P(X=2)=\frac{1}{6},P(X=3)=\frac{1}{6},P(X=4)=\frac{1}{6},P(X=5)=\frac{1}{6},P(X=6)=\frac{1}{6}$
In a discrete probability distribution, the probabilities assigned to each value must satisfy two conditions: (1) each probability must be between 0 and 1 (inclusive), and (2) the sum of all probabilities must equal 1.
**Continuous Probability Distribution:**
In contrast to discrete probability distributions, continuous probability distributions are associated with continuous random variables. Continuous random variables can take on an uncountably infinite number of values within a specified range or interval.
To provide an example, let's consider the normal distribution, also known as the Gaussian distribution. It is a commonly encountered continuous probability distribution. The normal distribution is defined by its mean ($\mu$) and standard deviation ($\sigma$). The probability of a random variable falling within a certain range is given by the area under the curve of the normal distribution within that range.
The probability density function (PDF) of the normal distribution is given by the following formula:
$P(a\le X\le b)={\int }_a^b\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}dx$
In this equation, $X$ represents the continuous random variable, and $a$ and $b$ represent the lower and upper bounds of the range.
Unlike discrete probability distributions, the probability of any single value in a continuous distribution is infinitesimally small because there are infinitely many possible values. Instead, the probabilities are expressed in terms of ranges or intervals.
In summary, the main difference between discrete and continuous probability distributions is the nature of the random variables they represent. Discrete distributions are associated with variables that can take on a finite or countably infinite set of values, while continuous distributions are associated with variables that can take on an uncountably infinite number of values within a specified range.