HallsofIvy said:Well, what have you tried? What are the mean and standard deviation of the uniform distribution over (0, 1)? What does the central limit theorem say about the distribution of the sum of samples?
The independence of the random variables allows for each variable to have its own unique probability distribution and outcomes, without being influenced by the other variables. This allows for a more accurate representation of the overall data and can simplify calculations and statistical analysis.
This notation indicates that there are 100 independent random variables, each with their own probability distribution. It is important to specify the range of i to accurately represent the data and to ensure that all variables are accounted for in the analysis.
The independence of random variables can be determined by examining the correlation between the variables. If there is little to no correlation, then the variables can be considered independent. Additionally, mathematical tests such as the Chi-Square test can be used to determine independence.
One example could be a study on the effects of different types of exercise on heart rate. Each participant in the study could be assigned a different type of exercise (i.e. running, weightlifting, swimming) and their heart rate could be measured independently. This would allow for a comparison of the effects of each type of exercise on heart rate without any external factors influencing the results.
One limitation is that it may not accurately represent real-life scenarios, as most events and variables in the real world are not truly independent. Additionally, if the variables are not chosen carefully, it can lead to biased or inaccurate results. It is important to carefully consider the use of independent random variables and their potential impact on the research findings.