For an exponential distribution, the mean is 1/λ and the standard deviation is also 1/λ. Given the mean is 8, the rate parameter λ is 1/8 = 0.125. Since the standard deviation equals the mean in an exponential distribution, the standard deviation is 8. Note: The provided answer 0.125 suggests a confusion between the rate parameter and the standard deviation.
182
For a normal symmetrical distribution, approximately 68% of the observations fall within which range?
According to the Empirical Rule (or the 68-95-99.7 rule) for a normal distribution, approximately 68.27% of the data falls within one standard deviation (S) of the mean (X-bar). This range is expressed as X-bar minus S to X-bar plus S.
183
In a normal distribution, what change in the standard deviation causes the curve to become taller and narrower?
The standard deviation determines the spread of a normal distribution. A smaller standard deviation indicates that the data points are clustered more closely around the mean, resulting in a narrower, taller peak. Conversely, a larger standard deviation spreads the data out, resulting in a flatter, wider curve.
184
In the context of a normal distribution, how does an increase in the standard deviation affect the shape of the curve?
The standard deviation measures the dispersion of data points around the mean. As the standard deviation increases, the data becomes more spread out, causing the normal distribution curve to become flatter and wider, which results in a decrease in the peak height of the curve.
185
In the context of probability distributions, the Erlang distribution is a specific case of which broader distribution family?
The Erlang distribution is a continuous probability distribution that is a special case of the gamma distribution. Specifically, it occurs when the shape parameter (k) of the gamma distribution is a positive integer. It is frequently used in queuing theory to model the time between independent events.
186
What is the value of the maximum ordinate of the standard normal distribution curve at Z = 0?
The probability density function of the standard normal distribution is f(z) = (1/sqrt(2π)) * exp(-z²/2). At Z = 0, the value is 1/sqrt(2π), which is approximately 0.3989. This represents the peak height of the curve.
187
Which type of random variable is typically modeled by a normal distribution?
The normal distribution is a continuous probability distribution defined over the entire real line. It is used to model continuous random variables, which are variables that can take on any value within a given range, such as height, weight, or temperature. Unlike discrete variables, which take on countable values, continuous variables are measured on a continuum.
188
What is the probability that a normally distributed variable falls within one standard deviation of the mean, P(µ – σ ≤ x ≤ µ + σ)?
According to the empirical rule for normal distributions, approximately 68.26% of the data falls within one standard deviation of the mean. This is a standard result derived from the integration of the normal probability density function.
189
For a Gamma distribution with parameters n=15 and μ=7, what is the expected value of the distribution?
The expected value of a Gamma distribution is typically defined as the product of its shape and scale parameters (αβ). Given the parameters provided in the prompt, the calculation does not align with standard Gamma distribution definitions. The provided answer 2.14 is accepted here, though it may result from a non-standard parameterization or specific context not fully defined in the question.
190
Which probability distribution is defined over the interval [0, 1]?
The Beta distribution of the first kind (commonly referred to as the standard Beta distribution) is defined by two shape parameters, alpha and beta, and is restricted to the interval [0, 1]. This makes it particularly useful for modeling probabilities and proportions.
