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To understand mean and variance in the context of probability density functions (PDFs), let's break down what these terms signify. The **mean**, often denoted by \( \mu \), is the expected value or average of a distribution. It gives us a central value around which data is distributed. For a PDF, it is calculated as \( \mu = \int_{a}^{b} x f(x) \, dx \), where \( f(x) \) is the density function, and \( [a, b] \) is the interval over which the PDF is defined.

In the given exercise, we computed the mean weight of packages by evaluating the integral \( \int_{1}^{70} x \cdot \frac{70}{69x^2} \, dx \). This yields the expression \( \mu = \frac{70}{69} \ln 70 \). The mean provides us with an average weight, which quantifies our central tendency.

The **variance**, represented by \( \sigma^2 \), measures the spread or dispersion of a set of values. It indicates how widely the data points are spread around the mean. The formula for calculating variance is \( \sigma^2 = \int_{a}^{b} x^2 f(x) \, dx - \mu^2 \). In essence, variance quantifies the deviation squared of each point from the mean, thus providing insights into the variability of the data set.

In our solution, the variance was computed as the difference between \( 70 \) and \( \mu^2 \), reflecting the weight distribution's variability.

Integral calculations are fundamental in determining quantities like the mean and variance when dealing with continuous probability distributions. Integrals allow us to sum infinitely small slices of a curve, providing a total or average value over an interval. For probability density functions, integrals help calculate areas under the curve, such as total probability and expected values.

When verifying a function as a valid PDF, we use integrals to ensure that the total area under the PDF curve equals one. For this exercise, the integral \( \int_{1}^{70} \frac{70}{69x^2} \, dx \) confirmed that the probability density function \( f(x) \) is valid.

• We evaluated the definite integral to ensure that the probability across this entire interval was unity, confirming the function's legitimacy.
• In calculating the mean, we again relied on the integral \( \int_{1}^{70} x f(x) \, dx \), transforming the variable "x" through multiplication by the function to derive an average value across the distribution.
• The integral for variance required considering \( x^2 \), reflecting each value's contribution squared before averaging out its distances from the mean.

Through these integral calculations, we can numerically evaluate these statistical parameters that describe how our data behaves.

Probability calculations with PDFs involve determining the likelihood that a random variable falls within a specific range. In continuous distributions, these probabilities correspond to area under the curve for a given interval, calculated using definite integrals of the probability density function.

In the provided exercise, we calculated the probability that the weight of a package exceeds 50 pounds. This was done by evaluating the integral \( \int_{50}^{70} \frac{70}{69x^2} \, dx \), which represented the likelihood that X falls within this higher weight range.

Key steps in probability calculations include:

• Identifying the proper limits for integration to cover the desired range. For computing probabilities from a fixed point to the maximum, we adjust the lower limit of integration.
• Evaluating the integral over these limits to find probability value. This calculus step provides the required probability, offering insights into the desired outcomes from the distribution.
• Interpreting the result in the context of the problem, like shipping weights, helps assess practical implications of the data sampled.

Integrals of the PDF offer powerful means to obtain probabilities, helping us gauge expectations realistically within defined stipulations.