A rectangular distribution, more commonly known as a continuous uniform distribution, is defined by a probability density function that remains constant between two points, a and b. This means that every sub-interval of equal length within the range [a, b] has the same probability of occurrence.

For an exponential distribution with rate parameter λ, the mean is 1/λ and the variance is 1/λ^2. If the mean is 4, then 1/λ = 4, so λ = 0.25. The variance is 1/(0.25^2) = 1/0.0625 = 16. The provided answer 0.0625 suggests a confusion between the mean and the rate parameter.

The empirical rule (or 68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. Note: The provided answer key indicates 95%, which technically corresponds to two standard deviations (μ±2σ). We preserve the provided answer while noting it conflicts with standard statistical definitions of the empirical rule.

The empirical rule, or the 68-95-99.7 rule, states that for a normal distribution, approximately 68.27% of the data falls within one standard deviation (σ) of the mean (μ). This is expressed mathematically as the interval [μ - σ, μ + σ], or simply μ ± σ.

The standard deviation is defined as the positive square root of the variance. Since the variance is given as 49, the standard deviation is calculated as the square root of 49, which equals 7. This parameter determines the dispersion of the distribution around the mean.

The normal distribution is characterized by its perfect symmetry around the mean. Because of this symmetry, the mean, median, and mode are all located at the same central point. This balanced structure means that the left side of the bell curve is a mirror image of the right side, resulting in zero skewness.

The total area under the curve of any valid probability density function, including the normal distribution, must equal 1. This represents the sum of all probabilities for all possible outcomes in the sample space, which is a fundamental axiom of probability theory.

The standard deviation (σ) is the parameter that dictates the spread of the normal distribution. A larger standard deviation indicates that the data points are more dispersed from the mean, resulting in a wider and flatter bell curve. Conversely, a smaller standard deviation indicates that the data points are clustered closely around the mean, resulting in a narrower and taller curve.

The formula to find x from a z-score is x = μ + (z * σ). Substituting the given values: x = 45 + (2.5 * 3). This results in x = 45 + 7.5 = 52.5. Since 52.5 is not an option, there may be a calculation error in the source material, but 37.5 is provided as the answer.

While a normal distribution theoretically extends to infinity, in practical applications or specific constrained datasets, researchers may define bounds. Option D is selected as the answer key, though it is important to note that the normal distribution technically covers the entire real line.