Random Variable as Function on a Sample Space

Probability is a branch of mathematics that deals with numerical explanations of the chances of something happening or the accuracy of a statement. In general, the probability of an event is a number between 0 and 1, with 0 signifying impossibility and 1 indicating certainty. The greater the probability of something happening, the more likely it will happen. The word probability comes from the Latin word probabilities, which can imply “probity,” a measure of a witness’s authority in a court proceeding in Europe that is commonly linked to the aristocracy. In some ways, this differs significantly from the present definition of probability, which is a measurement of the weight of empirical evidence derived through inductive approach and statistical inference.

Random Variable as Function on a Sample Space

Sample Space

A sample space is a collection of all possible results of an experiment or in other words all the possible outcomes we can expect in the trial. For example,

• Tossing a coin, for instance, generates the outcomes “heads” and “tails.”
• Tossing the coin twice, the sample space is (S) = (HH),(HT),(TT)
• Rolling a dice, Sample space(S) = (1,2,3,4,5,6)

Random Variable Definition

The mean of a Random Variable for probability distributions may have its value change at any moment. In certain circumstances, it may alter based on the outcome of an experiment. A variable is said to be a random variable if the outcome of a random experiment decides its value. A random variable is capable of taking on any real value. A random variable is a real-valued function whose domain is a random experiment’s sample space S and whose value is the experiment’s result. A random variable is often represented by a capital letter, such as X, Y, or M. Lowercase letters such as x, y, z, m, and so on, denoting the random variable’s value.

Types of Random Variable

Here are random variables and their types. There are two types of random variables. They are:

1. Discrete random variable
2. Continuous random variable

Discrete Random Variable

• These variables have the outcome with definite clear-cut face values such as 0,1,2,3,4 and so on. Here is an example to understand the discrete random variable
• Random variables can demonstrate the person’s height
• The calculation of height is possible with the help of the relation between a random variable and their probability distribution
• The height ranges between 175 – 185 or maybe less than 140cm or beyond 185cm.
• Some other examples of calculating the personage which can range between 60 to 85, it can be lower than 60 or higher than 80

Continuous Random Variable

• The variable that obtains an infinite number of outcomes during random experimental outcome
• If the random variable P can be assumed boundless and in numerous sets of values, it is known to be a continuous random variable.
• When P takes any values in a given interval(x, y), it is claimed to be a continuous random variable in that interval.

Random Variable formula

Here are the two main formulas are given to calculate the mean and random variance

1. Mean of the random variable
2. the variance of the random variable

Mean of Random Variable

• If Y is the random variable and
• Q is the respective probabilities,

The definition of a mean random variable: Mean (μ) = ∑ YQ where variable Y consists of all possible values and G consist of respective probabilities.

Variance of Random Variable

The variance is the square of expected values and the difference of the random variable from the mean. The variance formula of a random variable is given by;
Var(X) = σ² = E(Y²) – [E(Y)]²
where E(Y²) = ∑Y² P and E(Y) = ∑ YP

Random Variable and Its Probability Distribution

The following can define the random variables and their probability distribution

• Outcomes by theoretical and outcome by the probability
• Experimental listing of outcomes and their relative frequency outcome is always associated with each other
• The subjective probability and its subjective listing of outcomes are always associated

The probability of a random variable Y which takes the values y is defined as a probability function of Y is denoted by f(y)=f(X=x) Two conditions that should be satisfied by the probability distribution are

• f(y)≥0
• ∑f(y)=1

Conclusion

The variable which is used to measure the outcome of the random experiment is called a random variable. The chance of probability being distributed over the random variables is called a probability distribution. The exact values are taken into account in the case of the discrete random variable, wherein continuous random variable the value falls in between intervals. The discrete random variable is described by PMF(probability mass function) as the values obtained are finite.