If X1...Xn are independent and identically distributed, how is the mean of these random variables described?

Study for the Society of Actuaries Exam P. Immerse in flashcards and multiple-choice questions, each with hints and explanations. Gear up for your exam success!

Multiple Choice

If X1...Xn are independent and identically distributed, how is the mean of these random variables described?

The statement regarding the mean of independent and identically distributed random variables is best described by the Central Limit Theorem. When you consider a sample of i.i.d. random variables X1, X2, ..., Xn, each with mean μ and variance σ², the mean of these random variables, denoted as \(\bar{X} = \frac{1}{n}(X_1 + X_2 + ... + X_n)\), will also have specific properties. The mean of the sample mean, \(\bar{X}\), is indeed equal to the population mean μ, which is a constant. However, what sets the correct answer apart is that the sample mean \(\bar{X}\) will be approximately normally distributed for sufficiently large n, with a variance that is adjusted to the size of the sample. This variance is given by \(\frac{σ^2}{n}\), demonstrating that as n increases, the variability of the sample mean decreases. Thus, the sample mean does not collectively syphon the properties of the individual means in a straightforward addition manner, but rather redistributes the variance among the sample size. Hence, the correct answer is descriptive of the characteristics of the distribution of sample means and how

The statement regarding the mean of independent and identically distributed random variables is best described by the Central Limit Theorem. When you consider a sample of i.i.d. random variables X1, X2, ..., Xn, each with mean μ and variance σ², the mean of these random variables, denoted as (\bar{X} = \frac{1}{n}(X_1 + X_2 + ... + X_n)), will also have specific properties.

The mean of the sample mean, (\bar{X}), is indeed equal to the population mean μ, which is a constant. However, what sets the correct answer apart is that the sample mean (\bar{X}) will be approximately normally distributed for sufficiently large n, with a variance that is adjusted to the size of the sample. This variance is given by (\frac{σ^2}{n}), demonstrating that as n increases, the variability of the sample mean decreases.

Thus, the sample mean does not collectively syphon the properties of the individual means in a straightforward addition manner, but rather redistributes the variance among the sample size. Hence, the correct answer is descriptive of the characteristics of the distribution of sample means and how

More practice questions