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The normal distribution, often called the bell curve due to its iconic shape, is a fundamental concept in statistics and probability. It describes data that tend to cluster around a mean or average value. When plotted on a graph, it forms a symmetrical curve with a peak at its mean and tails that taper off equally on either side.

Key characteristics of a normal distribution include:

• It is symmetric around the mean, meaning that the left and right sides of the curve are mirror images.
• The mean, median, and mode of a normally distributed data set are equal.
• The empirical rule, which states that about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

The standard normal distribution is a special case where the mean is zero and the standard deviation is one. This is denoted as \(N(0, 1)\). A common application of normal distributions is in determining probabilities and understanding the spread of data around a central value.

In the problem from the textbook, the variables \(X\) and \(Y\) are both normally distributed. This is important, as it allows us to utilize the properties of the normal distribution when calculating the probability distribution of their sum, \(Z = X + Y\).

A random variable is a numerical outcomes of a random phenomenon. It assigns numbers to each outcome of a random process, and can be classified as either discrete or continuous.

• Discrete random variables take on a countable number of distinct values, like the roll of a dice.
• Continuous random variables, like the ones in this problem, can take on an infinite number of possible values within a given range.

In the context of the exercise, both \(X\) and \(Y\) are continuous random variables drawn from normal distributions, implying they can take any real value. Because they are independent, the probabilities concerning \(X\) do not affect those concerning \(Y\).

Therefore, to find the probability distribution of their sum \(Z = X + Y\), we leverage the property that the sum of two independent normally distributed random variables is also normally distributed. Knowing this is fundamental in dealing with problems of statistical sums like the one given in the textbook.

A probability density function (PDF) defines the likelihood of a random variable to take on a particular value. For continuous random variables, such as those considered in this exercise, the PDF is vital for determining probabilities over intervals.

Unlike probabilities for discrete random variables, the probability of a continuous random variable taking on an exact value is always zero. Instead, we find probabilities for continuous variables across intervals by calculating the area under the PDF curve.

For a normal distribution, the PDF has a distinct mathematical form:

• It is written as \(f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}\).
• The parameters \(\mu\) and \(\sigma\) represent the mean and standard deviation, respectively.

In the textbook problem, the sum \(Z = X + Y\) is a random variable with its PDF \(f_Z(z) = \frac{1}{\sqrt{2\pi(\sigma_1^2 + \sigma_2^2)}} e^{-\frac{(z - (\mu_1 + \mu_2))^2}{2(\sigma_1^2 + \sigma_2^2)}}\). This describes the probability distribution of \(Z\), using the properties of the PDF of normal distributions and the results obtained by summing independent random variables.