Random Variable and Probability Distribution

How probabilities are distributed throughout a random variable’s values is referred to as its probability distribution. A probability mass function, denoted by the symbol f, characterises the probability distribution of a discrete random variable called x. (x). Random variable values can be retrieved by using this function. An integer random variable’s probability function may be constructed by satisfying these two requirements:

1. f(x) cannot be negative for any of the integer random variable’s values.
2. the sum of its individual probabilities must equal one.

Types of Random Variable

Three sorts of random variables were mentioned in the beginning:

• Discrete Random Variable
• Continuous Random Variable
• Mixed Type

Let’s take a look at the many types of variables and some real-world applications of each.

Discrete Random Variable

If a discrete random variable can only take the values 0, 1, 2, 3, 4, 5, and so on, it is a discrete random variable Every conceivable value in the random variable’s probability distribution is listed in the probability mass function.

In an analysis, a random variable can be used to represent the height of a person chosen at random. Random variables can be regarded as functions that relate height to one another. There are several ways of looking at a random variable’s probability distribution, such as the likelihood that the height falls between 175 cm and 185 cm or 145 cm and 180 centimetres. Age is a random variable as well. As long as it’s between the ages of 45 and 50, it might potentially be under 40 or above 50 years old.

Continuous Random Variable

If a numeric variable may take on the values a and b in any unit of measurement, it is said to be continuous. If X is capable of taking on an unlimited number of values, it is said to be continuous. If X is a continuous random variable, it can take any value inside the interval (a, b).

Continuous random variables refer to random variables that don’t vary over time. When comparing numbers with a low probability of occurring, there are no “gaps” in the data. Although these variables nearly never have a clearly defined value c, their values are likely to be confined within tiny periods of time.

Mixed Type

If the cumulative distribution function is neither discrete nor continuous in all places, the random variable is considered mixed. There are discrete and continuous random variables that make up this mixture of variables. Weighted averages of the CDFs for each individual variable are used to calculate the CDF in this scenario.

To find a mixed-type random variable, you can throw a coin and spin the spinner only if the coin lands on the head. For a “tail,” X equals 1, but for a “head,” X equals the spinner’s value, the same as in the preceding example. This random variable has a 50/50 probability of ending up with a value of 1. For example, in a range of values where half the probability is present, the probability is still half as high.

The discrete, singular, and totally continuous components of any probability distribution on the real line are all present. The decomposition theorem of Lebesgue has further information. Dedication. However, this collection may be large enough to be counted in the discrete component (like the set of all rational numbers).

The formula for Random Variable

You can calculate the average and standard deviation of a sample of random variables using this formula. As a result, we’ll go through the following two formulas:

• Mean of a random variable
• The variance of the random variables

Mean (μ) = ∑XP, where X is the random variable and P is the relative probability.

where X is a list of all the possible values and P is a list of how likely each value is.

The variance of a Random Variable: The variance of a random variable X shows how far it is from the mean value. The formula for a random variable’s variance is Var(X) = 2 = E(X₂) – [E(X)]. 2 where E(X2) equals X₂P and E(X) equals XP.

The Probability Distribution of a Random Variable

A random variable’s probability distribution shows how the probabilities are spread out throughout the possible values of the random variable’s values. F(x) is the probability mass function used to describe the probability distribution of discrete random variables. For each value of the random variable, this function returns the probability of that value. One of the two characteristics that must be met in order to build a probability function for a discrete random variable is that the probability function f(x) must be nonnegative for each random variable value.

A probability distribution can never perform anything other than these two things:

• f(x)≥0
• ∑f(x)=1

The following are the most significant probability distributions:

• The Binomial distribution.
• Probability distributions based on the Poisson model
• Bernoulli’s probability distribution
• A distribution with an exponential probability.
• Descriptive statistics

Conclusion

It is a number that indicates the outcome of a statistical experiment. There are only two possible values for a discrete random variable: a finite number of values or an infinite number of values. In contrast, any value along the real number line can be assigned to a continuous random variable. A person’s weight in kilogrammes (or pounds) would be a continuous random variable, but the number of automobiles sold at a dealership on a given day would be a discrete random variable.