The data comes from following website, I random select 150 observations, x variable is Age. Mean :37 Variance: 361 Standard deviation: ) Suggest an appropriate probability distribution to approximate the histogram Part one: The data originates from a specified website, where 150 observations were randomly selected. The variable of interest is Age, with a mean of 37, variance of 361, and a standard deviation of 19. Based on the description of the data distribution — notably its positive skewness and the decreasing probability trend as age increases — an appropriate probability distribution to model this data is the exponential distribution. The exponential distribution is well-suited to modeling positively skewed continuous data where the probability decreases exponentially as the variable increases. Part two: Utilizing the procedure akin to Gretzky’s example, the method involves fitting a probability density curve to the histogram, performing a goodness-of-fit test, and computing several characteristics of the fitted distribution, such as the probability density function (pdf), cumulative distribution function (cdf), empirical cdf, moment generating function (mgf), expected value, and variance. The goal is to assess how well the exponential distribution approximates the observed data and to interpret these results in context.
Paper For Above instruction The analysis of age data with a sample size of 150 observations, as obtained from the specified website, provides a compelling application for statistical distribution fitting. The reported measure of central tendency—a mean age of 37—along with a variance of 361 and a standard deviation of 19, indicates a distribution that is positively skewed, a common characteristic in age-related data. Such skewness suggests that the data are not symmetrical and tend to have a longer right tail, which naturally leads to considering distributions that can accommodate this asymmetry. Given the nature of the data and its attributes, the exponential distribution emerges as a suitable model. The exponential distribution is defined for positive continuous variables and is characterized by its decreasing probability density, which aligns with the observed data trend where probabilities decrease as age increases. Mathematically, the exponential distribution’s probability density function (pdf) is expressed as: f(x; λ) = λ e −λx