In a binomial distribution with parameters n (number of trials) and p (probability of success), the expected value or mean is calculated as the product of the number of trials and the probability of success, represented as E(X) = np. This represents the average number of successes expected over many repetitions of the experiment.
362
Given a binomial distribution with p = 0.60, n = 3, and a random variable x = 4, what is the calculated z-score?
The z-score is calculated as (x - μ) / σ. For a binomial distribution, μ = np = 3 * 0.6 = 1.8, and σ = sqrt(npq) = sqrt(3 * 0.6 * 0.4) = sqrt(0.72) ≈ 0.8485. Thus, z = (4 - 1.8) / 0.8485 ≈ 2.59. Note: x=4 is outside the range [0,3], but the calculation follows the standard formula.
363
Which characteristic is true for the Poisson probability distribution?
A defining property of the Poisson distribution is that its mean (λ) is equal to its variance (σ² = λ). This distribution models the number of events occurring within a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event.
364
In the context of a binomial distribution, what is the sum of the probability of success (p) and the probability of failure (q)?
In a binomial experiment, each trial has exactly two mutually exclusive outcomes: success or failure. By definition, the sum of the probabilities of all possible outcomes in a sample space must equal 1. Therefore, p + q = 1.
365
Which specific parameter determines the shape of a Poisson distribution?
The Poisson distribution is defined by a single parameter, often denoted as lambda (λ). This parameter represents the average rate of occurrence for events in a fixed interval of time or space. Because the mean and variance of a Poisson distribution are both equal to λ, the parameter fully dictates the distribution's shape and spread.
366
What is the formula for calculating the standard deviation of a binomial distribution with parameters n and p?
For a binomial distribution, the variance is defined as npq, where n is the number of trials, p is the probability of success, and q is the probability of failure (1-p). The standard deviation is the square root of the variance, which results in the square root of npq.
367
Which discrete probability distribution is used to model the number of occurrences of an event within a small, fixed interval of time or space?
The Poisson distribution is specifically designed to model the number of times an event occurs in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. It is a fundamental discrete distribution for count data in statistics.
368
How is a binomial distribution characterized when the probability of success, p, is less than 0.5?
In a binomial distribution, if the probability of success p is less than 0.5, the distribution is positively skewed, meaning it is skewed to the right. Conversely, if p is greater than 0.5, the distribution is negatively skewed (skewed to the left). If p equals 0.5, the distribution is symmetric.
369
If an experiment involving 5 dice is performed 96 times, what does the value N represent?
In the context of frequency distributions for binomial experiments, N typically denotes the total number of repetitions or trials of the entire experiment. Here, the experiment of rolling 5 dice is repeated 96 times, making N equal to 96.
370
In a binomial distribution where n=6 and p=0.9, what is the value of P(X>6)?
In a binomial distribution with n=6, the random variable X can only take integer values from 0 to 6. Therefore, the probability of X being greater than 6 is exactly 0. Since 0 is less than 1, the statement 'less than one' is mathematically true, although 'zero' would be the most precise description of the probability.
