Types of Probability Distribution

A random variable’s probability distribution defines how probabilities are distributed throughout its values. A probability mass function, denoted by f, describes the probability distribution of a discrete random variable, x. (x). This function returns the probability of each random variable value being true. To construct the probability function for a discrete random variable, two requirements must be met: (1) f(x) must be nonnegative for each value of the random variable, and (2) the overall probability for each value of the random variable must equal one.

Types of Random Variable 

As stated in the introduction, there are two types of random variables:

• Binomial Random Variable

• Normal Random Variable

Let’s take a deeper look at the different sorts of variables and some instances.

Binomial Random Variable 

A discrete random variable has a finite number of possible values, such as 0, 1, 2, 3, 4, and so on. The probability mass function compares each of the possible values in a random variable’s probability distribution to a list of probabilities.

Allow a person to be picked at random and a random variable to represent the individual’s height in an analysis. The random variable may be rationally stated as a function that ties a person’s height to themselves. In terms of the random variable, it is a probability distribution that allows the probability that the height falls into any subset of plausible values to be calculated, such as the likelihood that the height falls between 175 and 185 cm or the chance that the height falls between 145 and 180 cm. The person’s age, which might vary from 45 to 50 years old, could be less than 40 or larger than 50, which is another random variable.

Normal Random Variable

If a numerically valued variable may take on the values a and b in any unit of measurement, it is termed continuous. If the random variable X may take on an infinite and uncountable set of values, it is termed continuous. When X takes any value inside that period, it is termed a continuous random variable (a, b).

A continuous random variable has a cumulative distribution function that is constant throughout. There are no “gaps” between the numbers that can be compared to those that are unlikely to materialise. Alternatively, these variables almost never take on an absolutely defined value c, but their value is likely to fluctuate in tiny intervals.

Difference between Binomial and Normal Distribution

1) The primary distinction between the binomial and normal distributions is that the former is a discrete distribution, whilst the latter is a continuous distribution.

This implies that a binomial random variable can only accept integer values like 1, 2, 3, and so on, but a regular random variable can take any real number value like 1.2 or 2.314, for example.

2) The second distinction is that a binomial random variable has a finite range, whereas a normal random variable has an infinite range.

A binomial random variable can only have a finite number of values, such as 1, 2,…., n. A normal random variable, on the other hand, can take any value between minus infinity and plus infinity, hence its range is unlimited.

3) The binomial distribution’s uses are restricted. It’s only used when there are just two possible outcomes for a trial: success or failure. We use the binomial distribution to compute probabilities when tossing a coin several times (because there are only two possibilities – heads or tails).

The normal distribution, on the other hand, has several uses in real-life scenarios, such as modelling a population’s height or weight distribution. Using the normal approximation to the binomial technique, the normal distribution may be used to derive probabilities for binomial distributions.

The formula for Random Variables

For a given set of data, the formula determines the mean and variance random variables. As a result, two important formulae will be defined here:

• Random variable mean

• Random variable variance

A random variable’s mean is defined as Mean (μ) = ∑XP where X is the random variable and P is the probability distribution.

where X stands for all conceivable values and P stands for their relative likelihood.

Random Variables and Probability Distribution

A random variable’s probability distribution

• A theoretical enumeration of possible outcomes and their probability

• An experimental table with the observed relative frequency of the outcomes.

• A subjective list of probable outcomes is accompanied by subjective probability.

f (x) = f (X = x) is the probability function of a random variable X that takes the values x.

Two conditions must always be met by a probability distribution

• f(x)≥0

• ∑f(x)=1

The most significant probability distributions are as follows

• Binomial probability distribution

• Probability distribution Poisson

• The probability distribution of Bernoulli

• The probability distribution that is exponential

• Descriptive statistics

Conclusion

A random variable represents the outcome of a statistical experiment numerically. A discrete random variable can only take one of two values: a finite number of values or an infinite series of values, but a continuous random variable can take any value along the real number line. For example, a random variable representing a person’s weight in kilograms (or pounds) would be discrete, but a random variable expressing a person’s weight in kilograms (or pounds) would be continuous.