The Poisson distribution is positively skewed for small values of lambda. However, as lambda increases, the distribution approaches a normal distribution, which is symmetric and mesokurtic. The provided answer reflects this limiting behavior.

A random variable that counts the number of successes in a fixed number of independent Bernoulli trials is defined as a binomial random variable. It is a discrete random variable, not a continuous one, as it can only take on integer values from 0 to n.

A binomial experiment consists of a fixed number of trials, each with two possible outcomes: success or failure. A fundamental assumption is that the outcome of any single trial does not influence the outcome of any other trial. Therefore, the trials are statistically independent, meaning the probability of success remains unchanged throughout the experiment.

In a binomial distribution, the mean (expected value) is defined as the product of the number of independent trials (n) and the probability of success in each trial (p). This formula, E(X) = np, represents the average number of successes expected over many repetitions of the experiment. Other options provided relate to variance or standard deviation, which are not the mean.

Independence of trials is a fundamental requirement for the binomial distribution. This means the outcome of any single trial must not influence the probability of success or failure in any subsequent trial. If trials are dependent, the binomial model is inappropriate, and other distributions, such as the hypergeometric distribution, must be considered.

In binomial experiments, the probability of success in a single trial is denoted by 'p', and the probability of failure is denoted by 'q'. Since the outcomes are binary, they satisfy the condition p + q = 1.

The mean of a binomial distribution is calculated as μ = np, where n is the number of trials and p is the probability of success. Given μ = 857.6 and p = 0.64, we solve for n: n = 857.6 / 0.64 = 1340.

Given mean np = 6 and variance npq = 2.4. Dividing variance by mean gives q = 2.4 / 6 = 0.4. Since p = 1 - q, then p = 0.6 or 3/5. Substituting p back into the mean formula: n * 0.6 = 6, which yields n = 10. Thus, the parameters are n = 10 and p = 0.6.

In a negative binomial distribution, where r represents the number of successes and p is the probability of success, the mean is defined as r/p. This distribution models the number of trials required to achieve a specified number of successes in a sequence of independent Bernoulli trials.

Independence is a core assumption of the binomial distribution. It implies that the outcome of any single trial does not affect the probability of success or failure in any other trial. This independence allows for the multiplication of probabilities, which is necessary to derive the binomial probability mass function.