Final answer:
The characteristic function of the standard normal distribution is e-(t2/2). For practical calculations concerning probabilities, graphing calculators' functions like normalcdf and invNorm are used to find areas under the normal curve and corresponding z-scores.
Step-by-step explanation:
The characteristic function of a standard normal distribution, which is denoted as Z ~ N(0, 1), plays an important role in probability theory and statistics. The characteristic function, φ(t) for a standard normal distribution, is e-(t2/2). To compute this, one would integrate the function eitx against the standard normal probability density function.
In a practical context, if we want to know the cumulative probability associated with a particular value, we use the function normalcdf on a graphing calculator such as the TI-83, 83+, or TI-84+. To find a z-score that corresponds to a given cumulative probability, we can use the invNorm function. For instance, using invNorm(0.95, 0, 1) would find the z-score for which 95% of the distribution lies to the left.