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Understanding the probability density function (PDF) is crucial when studying continuous random variables. A PDF represents how the values of a random variable are distributed; it is the graph of a function that maps all possible values of the random variable to their corresponding probabilities. For the standard normal distribution, the PDF is given by the formula:
\[ f(z) = \frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}} \]
In simpler terms, this function tells us the probability of a random variable (in this case, the standard normal variable) falling within a particular range. The formula illustrates that the probability smoothly changes across different values of z rather than jumping from one probability to another.
Fundamentally, the PDF shows that the probabilities are highest near the mean, which is 0 for standard normal distribution, and taper off as we move away from the center. Importantly, the total area under the PDF curve equals 1, which corresponds to the certainty that a random variable will take some value within its range.

A z-score signifies how many standard deviations a data point is from the mean. When a random variable follows a standard normal distribution, any value of it can be converted into a z-score. The standard normal distribution's mean \(\mu = 0\)\ and standard deviation \(\sigma = 1\)\ are the benchmarks for these scores. The z-score formula is:
\[ z = \frac{X - \mu}{\sigma} \]
In practice, a z-score also serves as a tool for finding probabilities associated with the standard normal distribution. For example, to find the probability that the random variable takes a value greater than any particular z-score, one would subtract the cumulative probability up to that z-score from the total probability, 1. Moreover, z-scores help in standardizing different data sets, enabling comparisons across various scales and distributions.
Returning to the exercise, to calculate \(P(z > 1.16)\), we need to find the cumulative probability up to z-score 1.16 and deduct it from 1. This step connects the abstract idea of a z-score to a tangible method for determining probabilities.

The term normal random variable refers to a variable that assumes values guided by the normal distribution, which is characterized by its bell-shaped curve. When the mean \(\mu\) and standard deviation \(\sigma\) of this distribution are 0 and 1, respectively, we call it a standard normal distribution. The significance of a standard normal random variable lies in its predictability; it shows us how frequently or rarely specific values occur.
Standard normal random variables are valuable in statistics because they allow us to evaluate and interpret various statistical outcomes. They are commonly used to model uncertainties and help gauge standard deviations from the mean in natural or social phenomena.
With a robust understanding of a normal random variable, one can apply the principles to more complex statistics, including hypothesis testing and confidence intervals. Moreover, since it's standardized, any normally distributed variable with a different mean or standard deviation can be transformed into a standard normal random variable using z-scores, thus simplifying calculations of probabilities and other statistical measures.