For a binomial distribution, the kurtosis is given by the formula [1 - 6p(1-p)] / [np(1-p)]. With n=7, p=0.3, and q=0.7, the variance npq = 7*0.3*0.7 = 1.47. The numerator is 1 - 6(0.21) = 1 - 1.26 = -0.26. The kurtosis is -0.26 / 1.47, which is approximately -0.1768. The provided answer 0.18 is the absolute value of this result.
342
In a binomial experiment, how is the number of trials characterized?
A fundamental requirement of a binomial experiment is that the number of trials (n) must be fixed in advance. Each trial must be independent, have only two possible outcomes (success or failure), and maintain a constant probability of success.
343
Under what condition are the mean, median, and mode of a binomial distribution identical?
A binomial distribution is perfectly symmetric when the probability of success (p) is 0.5. In any symmetric unimodal distribution, the mean, median, and mode coincide at the center of the distribution. When p = 0.5, the distribution of successes is balanced, resulting in equal values for these three measures of central tendency.
344
Given a Poisson distribution parameter λ = 9 and a random variable x = 5, calculate the z-score.
For a Poisson distribution, the mean is λ and the variance is also λ. The standard deviation is the square root of λ. Here, mean = 9 and standard deviation = √9 = 3. The z-score is calculated as (x - mean) / standard deviation = (5 - 9) / 3 = -4 / 3 ≈ -1.333. The closest option is -1.34.
345
In the context of a binomial distribution, how is the probability of failure represented?
In a binomial distribution, each trial results in either success (p) or failure (q). Since these are the only two mutually exclusive and exhaustive outcomes, their probabilities must sum to one. Therefore, the probability of failure is defined as q = 1 - p.
346
Calculate the mean of a binomial distribution where the probability of success p = 0.1 and the number of trials n = 400.
The mean (expected value) of a binomial distribution is calculated using the formula E(X) = n * p. Given n = 400 and p = 0.1, the calculation is 400 * 0.1 = 40. Therefore, the mean of this distribution is 40.
347
In a binomial probability distribution, how does the probability of success change as the parameter 'p' increases?
The binomial distribution is defined by the parameters n (number of trials) and p (probability of success in a single trial). By definition, the probability of success in the distribution is directly tied to the value of p. As the value of p increases, the likelihood of achieving a success in any given trial increases, thereby shifting the distribution's mean toward a higher number of successes.
348
Given a distribution with a success probability of 0.68 and 4 trials, what is the mean of the Poisson probability distribution?
The mean (λ) of a Poisson distribution is calculated as n * p. Here, 4 * 0.68 = 2.72. While the question mentions a Poisson distribution, this calculation is technically the mean of a Binomial distribution. We retain the answer as provided in the source.
349
Which probability distribution is typically used to model the number of independent events occurring within a fixed interval of time or space?
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. It is applicable when events occur independently at a constant average rate. This distribution is widely used in fields like telecommunications, insurance, and quality control to model count data where the occurrence of one event does not affect the probability of another.
350
What is the general expression for the binomial frequency distribution when an experiment is repeated N times?
The binomial frequency distribution is obtained by multiplying the total number of experiments (N) by the probability mass function of the binomial distribution. The formula is N * C(n, x) * p^x * q^(n-x), where C(n, x) is the binomial coefficient.
