A binomial distribution is negatively skewed when the probability of success (p) is greater than 0.5. Since p = 0.6, the distribution is skewed to the left, meaning the tail of the distribution extends toward the lower values. This reflects a higher concentration of probability mass at the higher end of the range.

In a binomial distribution, the skewness is determined by the probability of success 'p'. When p > 0.5, the distribution has a longer tail on the left side, indicating negative skewness. Conversely, if p < 0.5, the distribution is positively skewed (skewed to the right). If p = 0.5, the distribution is perfectly symmetric.

The variance of a binomial distribution is defined by the formula Var(X) = np(1-p). Given n = 20, p = 0.6, and q = 1-p = 0.4, the calculation is 20 * 0.6 * 0.4 = 4.8. This measure quantifies the dispersion of the random variable around its mean.

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. The mean (expected value) of this distribution is determined by multiplying the total number of trials (n) by the probability of success in a single trial (p). Thus, the formula is μ = np.

In a binomial distribution, the mean is np and the variance is npq, where q = 1-p. Since p is a probability between 0 and 1, q must also be between 0 and 1. Multiplying the mean (np) by a value q < 1 results in a variance that is strictly less than the mean, provided 0 < p < 1.

The variance of a binomial distribution is given by the formula npq. Since p + q = 1, we can write the variance as np(1-p). This quadratic function reaches its maximum value when p = 0.5, which implies q = 0.5. Thus, the variance is maximized when the probability of success and failure are equal.

When the parameter lambda of a Poisson distribution is an integer, the probability mass function takes the same maximum value at both k = lambda and k = lambda - 1. This results in two modes, making the distribution bimodal. If lambda is not an integer, the distribution is unimodal with the mode at the integer part of lambda. This is a specific property of the Poisson distribution.

In a binomial distribution, if the probability of success (p) equals the probability of failure (q), which is 0.5, the distribution is perfectly symmetric. This occurs because the likelihood of observing a certain number of successes is identical to the likelihood of observing the corresponding number of failures, resulting in a balanced distribution.

A core assumption of the binomial distribution is that the probability of success (p) is constant for every independent trial. If the probability of success were to change from trial to trial, the process would no longer follow a binomial distribution, but rather a more complex model like the Poisson binomial distribution.

For a binomial distribution, the mean is calculated as the product of the number of trials (n) and the probability of success (p). Here, n = 3500 and p = 0.55. Multiplying these values (3500 * 0.55) yields 1925, which represents the expected number of defective items.