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In probability theory, independent random variables are those variables whose outcomes do not influence each other. This means that the occurrence of an event associated with one variable does not affect the probability of an event associated with another variable. For instance, if we look at our exercise, we have two independent gamma distributed random variables: the dead load, denoted as \(X_1\), and the occupancy load, denoted as \(X_2\).
The independence of these variables implies that the total sustained load, which is the sum of these two variables, is simply the sum of their individual loads. This independence simplifies calculations because the mean and variance of the total load \(Y\) can be separately considered as the sum of their individual means and variances.

• For independent events, the joint probability is the product of their probabilities.
• Calculations involving independent random variables are often simplified due to their non-related outcomes.

This property is fundamental as it allows us to calculate complex scenarios in real-world applications efficiently.

The mean and variance are essential characteristics of a probability distribution, providing insights into its central tendency and spread, respectively. For a gamma distribution specified by parameters \(\alpha\) and \(\beta\), the mean is given by \(E[X] = \alpha \beta\) and the variance by \(Var[X] = \alpha \beta^2\).

• The mean provides a measure of the central or ‘average’ value of the distribution.
• The variance indicates how much variability there is from the mean.

In the given exercise, we calculated the mean and variance for two gamma-distributed loads, \(X_1\) with parameters \(\alpha_1 = 50\), \(\beta_1 = 2\) and \(X_2\) with parameters \(\alpha_2 = 20\), \(\beta_2 = 2\). For \(X_1\), the mean and variance are 100 kips and 200 kips², respectively. For \(X_2\), they are 40 kips and 80 kips². Subsequently, the total load \(Y = X_1 + X_2\) has a mean of 140 kips and a variance of 280 kips². The total load inherits these properties through linear combinations which are greatly facilitated due to independence.

Quantile calculation is a method used to understand the probability distribution better by identifying specific points of the data. In our case, we needed to find the 15/16 quantile of the gamma distribution of the total load \(Y\). This entails finding the load value that is exceeded only 1/16 of the time, or equivalently, not exceeded 15/16 of the time.

• The quantile indicates the probability that a value will occur below a certain point in the distribution.
• To find quantiles, practitioners often use statistical tables or software that provide cumulative distribution values.

For the gamma distribution with parameters \(\alpha_Y=70\) and \(\beta_Y=2\), this process involves computational methods as analytic solutions might not be straightforward. Therefore, flipping through the quantile tables or utilizing statistical software is common practice to find such critical values.

Probability distribution properties are key attributes that define the behavior of the distribution. For the gamma distribution, these include properties such as the shape, scale, and summation properties. The gamma distribution is characterized by its shape parameter \(\alpha\) and scale parameter \(\beta\).

• The shape parameter influences the skewness and the way the probability mass is distributed.
• The scale parameter impacts the spread or variability of the distribution.

In the context of our exercise, both \(X_1\) and \(X_2\) have gamma distributions, and their sum \(Y\) naturally follows a gamma distribution as well, owing to their independent nature. The overall properties of \(Y\) include aggregated shape parameter \(\alpha_Y = \alpha_1 + \alpha_2\) and a common scale parameter \(\beta_Y\), giving it a new distribution but with predictable characteristics because of its origins. These properties are crucial for understanding how random events aggregate and influence the system's behavior.