The median of a normal distribution is equal to its mean (µ). In the context of Z-scores, the mean is transformed to 0. While the answer key provided is C, note that the Z-score of the median is mathematically 0. The choice C is technically incorrect as it refers to the raw score, not the Z-score.

In a Gamma distribution with shape parameter alpha and scale parameter beta, the mean is given by alpha times beta and the variance is given by alpha times beta squared. If beta equals 1, the mean becomes alpha and the variance becomes alpha, thus making them equal. This property is a characteristic feature of the Gamma distribution under the specified parameter condition.

The Beta distribution of the second kind, also known as the Beta prime distribution, has a variance formula that depends on its two shape parameters. The provided expression corresponds to the theoretical variance of this specific distribution type.

A z-score, or z-statistic, represents the number of standard deviations a data point is from the mean of a distribution. It is the result of standardizing a normal random variable by subtracting the mean and dividing by the standard deviation, resulting in a standard normal distribution with mean 0 and variance 1.

The standard normal distribution is centered at a mean of zero. By definition, the z-score represents the number of standard deviations a value is from the mean. Values located to the right of the mean are greater than the mean, resulting in positive z-scores, while values to the left are less than the mean, resulting in negative z-scores.

The exponential distribution is defined for non-negative random variables. The rate parameter (λ) must be positive, and since the mean (μ) of an exponential distribution is defined as 1/λ, the mean must also be strictly greater than zero to ensure the distribution is valid.

For a Gamma distribution with shape parameter 'a' and rate parameter 'b', the probability density function is proportional to x^(a-1) * exp(-bx). The mode of this distribution is found by maximizing the density function, which results in the value (a-1)/b, provided that a is greater than or equal to 1.

For a Gamma distribution with shape parameter alpha and scale parameter beta, the variance is defined as alpha multiplied by the square of beta. This relationship is a fundamental property used in characterizing the spread of Gamma-distributed random variables.

The normal distribution is a symmetric probability distribution. Because the total area under the curve is equal to 1, the vertical line passing through the mean divides the area into two equal halves, each representing 0.5 or 50% of the total probability.

The standard deviation determines the spread of a normal distribution. A larger standard deviation indicates that the data points are more dispersed from the mean, which results in a wider and flatter bell-shaped curve. Conversely, a smaller standard deviation results in a taller, narrower peak.