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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. In this article, we will discuss sample space & random variables.
Sample space
The set of every possible outcome obtained from a random experiment is known as sample space. It is represented by the S symbol.
For example:
Tossing the coin twice, the sample space is (S) = (HH), (HT), (TT)
Rolling a dice, sample space (S) = (1, 2, 3, 4, 5, 6)
Day in the months, S (Sample Space) = {Jan, Feb, March, April, May, June, July, August, Sept, Oct, Nov, Dec ❵
Days of the week (S)= {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
Random variable
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, denote the random variable’s value.
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.
Types of random variables
Here are random variables and their types. There are two types of random variables. They are
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 its probability distribution. The average height ranges between 175 – 185 cm or maybe less than 140cm or beyond 185cm.
As another example, calculating a person’s age which can range between 60 to 85, it can be lower than 60 or higher than 80.
Continuous random variable
If the variable obtains an infinite number of outcomes during random experimental outcome is termed continuous random variable. 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 formulas
The two main formulas calculate the mean and random variance.
Mean of a random variable
If Y is the random variable and Q is the respective probabilities, the formula for the mean random variable is:
Mean (μ) = ∑ YQ
Variance of a 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 variables and their 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 and 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 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. In the 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.
