Normal Distribution Probability Problems

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where the mean, median, and mode are all equal.

The parameters of a normal distribution are the mean (μ) and the standard deviation (σ).

The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

The inverse normal function is used to find the value of a variable given a specific percentile in a normal distribution.

The cumulative distribution function (CDF) represents the probability that a random variable takes on a value less than or equal to a specific value.

The binomial probability formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

Percentiles indicate the relative standing of a value within a distribution, showing the percentage of data points that fall below that value.

In a normal distribution, the probability of a value being less than the mean is 0.5 or 50%.