Financial Risk Manager Part 1

The Central Limit Theorem (CLT) is a fundamental theorem in statistics that describes the shape of the distribution of sample means. It states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will approach a normal distribution with a mean (μ) and standard deviation (σ/√n). This will hold true regardless of the shape of the population distribution. The key condition is that the sample size should be sufficiently large, typically n > 30. This theorem forms the basis of many statistical procedures and concepts, including confidence intervals and hypothesis testing.

Why Choice A is incorrect: While it is true that the sum of independent and identically distributed (i.i.d.) random variables tends to be normally distributed when the sample size is large, this statement does not fully encapsulate the essence of the Central Limit Theorem (CLT). The CLT also specifies that the mean of these sums will approach the population mean and their variance will approach σ²/n, which are not mentioned in this choice.

Why Choice B is incorrect: This option only talks about how the sum of i.i.d random variables approaches a normal distribution as n becomes large. However, it fails to mention anything about how their mean and variance behave, which are crucial aspects of CLT.

Why Choice C is incorrect: This choice incorrectly states that it's the sampling distribution itself that approaches a normal distribution with mean μ and variance σ²/n as n becomes large. In reality, according to CLT, it's specifically the sampling distribution of sample means that exhibits this behavior.