Question

Familiarity with random variables is essential to understand the basics of portfolio theory. Please review and explain the significance of basic concepts about random variables, namely, the mean, the variance, the standard deviation, and the correlation.

Provide your explanations and definitions in detail. Provide references for content when necessary. Support your statements with peer-reviewed in-text citation(s) and reference(s).

Answer 1

In Statistics or data analysis, the various descriptive or summary statistics on a particular random variable essentially signifies the overall statistical nature, characteristics, and overview of the variable which is extremely essential for a meaningful and comprehensible statistical understanding and analysis of the variable. In this context, the mean of the average of any random variable is highly important and useful to identify the central tendency or measure of the data on the particular variable which is calculated by adding or summing all the individual data or observations of the variable and then dividing by the total number of the data or the observations. Therefore, the mean or the statistics average of any variable encompasses each individual data or observation related to any variable and is perhaps the most comprehensive measure of the central measure of tendency of the entire dataset. The mean or the average also provides an overall idea about the summary of the entire dataset pertaining to any particular variable as it reflects the particular point or level at which most of the observations or data of the variable is centered around or concentrated. Now, the variance of any variable basically shows the overall spread or stretch of any particular dataset or the overall deviation of the individual data or observations from the measure of central tendency such as mean or average. It is mathematically calculated as the average squared deviation of each individual data or observation of the variable from the mean or average of the dataset and hence, provides an underatdning or estimation about the overall spread or extent of the entire dataset of any variable. Therefore, as a descriptive statistics it is a measure of the data variability which can ideally provide an idea about how spread out or concentrated the individual observations or the data within any variable is. The higher the value of the variance, the higher the spread of any dataset and vise versa.

The standard deviation is mathematically computed to derived as the square root of the variance and compared to the variance, it is a more common measure of the spread or the variability of any dataset related to any random variable. It essentially shows how much spread out each individual data or observations are from the measure of the central tendency such as mean or average or is a good indicator of any statistical outlier in the entire data set. An outlier in the data set basically refers to any particular observation or data that is an exception and is not following the common trend of the data or observations that are mostly clustered or concentrated around the mean or average and is located far away from these data or observations. Hence, standard deviation can be extremely useful in the identification of any outlier/s within any dataset that might affect or influence the overall spread or variability of the dataset regarding any variable. Now, statstictical correlation usually measures the statistical relationship or association between any two or multiple random variables. It basically reflects how strongly this variables are statistically related or associated with each other. The mathematical value or coefficient of the statistical correlation can be either negative, zero, and positive implying that the concerned variables are negatively related or associated, no relation or association between the variables, and the variables are positively related or associated respectively. Hence, te statistical correlation between the random variables essentially shows how much of the variability of one variables is statistically associated with the variability of another and it can be practically useful in undertaking predictive decision-making as an increase or decrease in the value of any variable can be explained by the variability of the other based on the measure or coefficient of the correlation. Therefore, it can also be used to control or manipulate the variation in one variable by adjusting the variation in the other on the basis of the related correlation value or measure, as part of any rational and predictive decision-making process. For example, if household and individual consumption level on various goods and services in any country is positively correlated with the respective household/individual disposable income, then in order to increase the overall consumption level in the country the government can implement policies or measures to enhance the income level of the households and inidividuals.

  References

https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214891-eng.htm

https://statistics.laerd.com/statistical-guides/descriptive-inferential-statistics.php#:~:text=Descriptive%20statistics%20are%20very%20important,simpler%20interpretation%20of%20the%20data.

https://www.sciencedirect.com/science/article/pii/S0104001417300167

https://socratic.org/questions/what-significance-does-variance-hold