Final answer:
The random variable b has the same distribution as the sum of 8 independent random variables because it represents the total number of successes in 8 independent trials, making it a binomial distribution. Additionally, due to the central limit theorem, as the sample size (number of variables being summed) increases, the distribution tends to become normally distributed.
Step-by-step explanation:
The question is asking to explain why a random variable b has the same distribution as the sum of 8 independent random variables. To answer this, we need to understand the concept of the binomial distribution and the central limit theorem. A binomial distribution is described by X~B(n, p), where n is the number of trials, p is the probability of success in each trial, and X represents the total number of successes. If the random variable b represents the total number of successes in 8 independent trials, then it has a binomial distribution.
The central limit theorem states that as the size of samples increases, the distribution of the sum of those random variables tends to become normally distributed. This is true regardless of the original distribution of the individual random variables, assuming they are independent and identically distributed. Thus, if b is the sum of 8 such independent variables, its distribution would approximate a normal distribution with specific mean and standard deviation defined by μ = np and σ = √npq, respectively.
In the context of the given information, if we have a binomially distributed variable with parameters 8 and 560, the distribution of b as the sum of 8 of these would be similarly binomial with the same parameters, since the sum of binomial distributions with the same probability of success p is also a binomial distribution.