Mean of the Random Variable

Introduction

What is a random variable? A random variable is an unknown value or a function that assigns values to each of an experiment’s findings. Random variables are frequently denoted by letters and can be classed as discrete, with defined values, or continuous, which can have any value within a continuous range. Random variables are often employed in econometric or regression analysis to find statistical correlations.

The mean of the random variable X is a weighted average of the random variable’s potential values. Unlike the sample mean of a set of observations, which assigns equal weight to each observation, the mean of the random variable assigns weight to each result xᵢ based on its probability, pᵢ. This article, that is, the mean of the random variable study material, can answer all your questions regarding the topic.

How to Find Out Mean of the Random Variable

The commonly used symbol for the mean (also known as the expected value of X) is μ, which is officially defined as;

       = 

The mean of the random variable represents the variable’s long-run average, or the predicted average outcome across many observations.

To calculate the mean of the random variable;

• Multiply each value by its probability
• Add them up

Based on the law of large numbers, the observed random mean of the random variable from increasing observations will approach the distribution mean. As the observations increase, the mean of these data approaches the real mean of the random variable. However, this does not guarantee that short-term averages will reflect the mean.

Consider a random variable U with discrete uniform distribution with values 1,2,…,m. The mean is calculated as,

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       = 

      = 

      =   

For example, the mean of a fair die roll is    

The graph below depicts the probability curve for the outcome of rolling a fair die once more. This distribution is symmetric, and the mean 3.5 is in the middle; it is on the axis of symmetry.

A broad physical interpretation of the mean of the random variable   with  . Assume the x-axis is an infinite see-saw in either direction, with weights equal to   for each conceivable value x of X. The mean X is then at the point where the see-saw balances. In other words, it is located in the system’s centre of mass. If the distribution of a discrete random variable is represented visually, you should be able to approximate the value of its mean using the ‘centre of mass’ concept.

Properties of Mean

• When a random variable X is multiplied by the value b and added to the value a, the mean is modified as follows,  

• The sum of the means of two random variables, X and Y, is the mean of their sum,  

• If  ,  , … ,   are random variables , then   
• For random variables, X and Y,   Here, X and Y must be independent.
• If a is any constant and X is a random variable,  [aX] = a  [X] and  [X + a] =  [X] + a.
• For any random variable, X > 0,  (X) > 0.
• (Y) ≥  (X) if the random variables X and Y are such that Y ≥ X.

Let us take an example and understand it further,

Assume a person plays a gambling game in which she can lose $1.00, break-even, win $3.00, or win $10.00 each time she plays. The table below shows the probability distribution for each outcome:

Outcome        -$1.00        $0.00        $3.00        $5.00        

Probability          0.30         0.40         0.20         0.10        

The mean outcome for this game is calculated as follows:

 = (-1  .3) + (0  .4) + (3 .2) + (10  0.1) = -0.3 + 0.6 + 0.5 = 0.8.

In the long term, the player may anticipate to earn around 80 cents playing this game, indicating that the chances are in her favour.

Assume the casino in the preceding gambling game recognises it is losing money in the long run and chooses to alter the payment levels by deducting $1.00 from each reward. The new probability distribution is-

Outcome        -$2.00        -$1.00         $2.00         $4.00        

Probability          0.30          0.40          0.20          0.10        

The new mean is (-2 0.3) + (-1 0.4) + (2 0.2) + (4 0.1) = -0.6 + -0.4 + 0.4 + 0.4 = -0.2. This is similar to deducting $1.00 from the original mean value, 0.8 -1.00 = -0.2.

The casino might expect to gain 20 cents in the long term with the revised payments.

Conclusion

The Mean of the random variable is an important concept in probability and statistics. Assume you want to know or make an educated guess about your performance on the five maths examinations. The total marks for each of the tests are the same. What can you say about your performance based on the grades you received?

You may do so by taking the average of your grades and estimating your total performance. This average mark will tell you whose marks you are most similar to.

We can calculate the mean of the random variable using probability and statistics. The term average refers to the mean, expected value, or expectation in probability and statistics. The mean of the random variable indicates the central tendency of the variable. Refer to the study material notes on the mean of the random variable for any doubts.