Random Variables
Random Variable is a real-valued function whose domain is the sample space of the random experiment. It is represented as X(sample space) = Real number.
We need to learn the concept of Random Variables because sometimes we are just only interested in the probability of the event but also in the number of events associated with the random experiment. The importance of random variables can be better understood by the following example:
Why do we need Random Variables?
Let’s take an example of the coin flips. We’ll start with flipping a coin and finding out the probability. We’ll use H for ‘heads’ and T for ‘tails’.
So now we flip our coin 3 times, and we want to answer some questions.
- What is the probability of getting exactly 3 heads?
- What is the probability of getting less than 3 heads?
- What is the probability of getting more than 1 head?
Then our general way of writing would be:
- P(Probability of getting exactly 3 heads when we flip a coin 3 times)
- P(Probability of getting less than 3 heads when we flip a coin 3 times)
- P(Probability of getting more than 1 head when we flip a coin 3 times)
In a different scenario, suppose we are tossing two dice, and we are interested in knowing the probability of getting two numbers such that their sum is 6.
So, in both of these cases, we first need to know the number of times the desired event is obtained i.e. Random Variable X in sample space which would be then further used to compute the Probability P(X) of the event. Hence, Random Variables come to our rescue. First, let’s define what is random variable mathematically.
Random Variable Definition
Random Variable is a function that associates a real number with an event. It means assigning a value (real number) to every possible outcome. In more mathematical terms, it is a function from the sample space Ω to the real numbers. We can choose our random variable according to our needs.
A random variable is a real valued function whose domain is the sample space of a random experiment
To understand this concept in a lucid manner, let us consider the experiment of tossing a coin two times in succession.
The sample space of the experiment is S = {HH, HT, TH, TT}. Let’s define a random variable to count events of head or tails according to our need, let X is a random variable that denotes the number of heads obtained. For each outcome, its values are as given below:
X(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0.
More than one random variable can be defined in the same sample space. For example, let Y is a random variable denoting the number of heads minus the number of tails for each outcome of the above sample space S.
Y(HH) = 2-0 = 2; Y (HT) = 1-1 = 0; Y (TH) = 1-1= 0; Y (TT) = 0-2 = – 2
Thus, X and Y are two different random variables defined on the same sample.
Note: More than one event can map to same value of random variable.
Probability Distribution – Function, Formula, Table
A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.
Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.
A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.
In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.
Table of Content
- What is Probability Distribution?
- Probability Distribution Definition
- Random Variables
- Random Variable Definition
- Types of Random Variables in Probability Distribution
- Probability Distribution of a Random Variable
- Probability Distribution Formulas
- Expectation (Mean) and Variance of a Random Variable
- Expectation
- Variance
- Different Types of Probability Distributions
- Discrete Probability Distributions
- Bernoulli Trials and Binomial Distributions
- Binomial Distribution
- Cumulative Probability Distribution
- Probability Distribution Function
- Probability Distribution Table
- Prior Probability
- Posterior Probability
- Solved Questions on Probability Distribution