91Ó°ÊÓ

The normal distribution is a fundamental concept in statistics and probability theory, used to describe how data values are distributed. It's often referred to as a "bell curve" because of its bell-shaped appearance when graphed. Most of the data points will cluster around the mean or average value, with fewer and fewer points as you move away from the mean on either side.

The normal distribution is defined by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)). The mean is the center of the distribution, and the standard deviation measures the spread of the distribution. In our exercise, the speeds of both airplanes are normally distributed. The first plane has a mean speed of 520 km/hr, and the second plane has a mean speed of 500 km/hr, both with a standard deviation of 10 km/hr.

Understanding these parameters helps predict how likely airplanes or other objects will deviate from their average speeds, which is essential in practical scenarios like air traffic management.

In statistics, a random variable is a variable whose values are determined by the outcomes of a random process. Random variables can be discrete or continuous. For example, flipping a coin results in a discrete random variable, while measuring a person's height is a continuous random variable.

In this problem, the speed of each airplane is represented as a random variable, denoted as \(X\) for the first plane and \(Y\) for the second plane. These variables are continuous because the speed can take on any value within a given range, governed by the normal distribution parameters specified above.

Random variables enable us to calculate probabilities and make statistical inferences. In our scenario, examining \(X - Y\) gives us insights into the relative speeds of the two planes. The difference encapsulates whether one plane can catch up to another over a period.

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. This type of distribution is particularly useful because any normal distribution can be transformed into a standard normal distribution using standardization.Standardizing converts a normal random variable \(Z\) with any mean \(\mu\) and standard deviation \(\sigma\) into a version \(Z'\) as follows:\[Z' = \frac{Z - \mu}{\sigma}\]In our problem, we calculate \(Z = X - Y\), which is normally distributed with mean 20 and variance 200. We then transform \(Z\) to \(Z'\) such that we can use the standard normal distribution tables to find probabilities more easily.

By doing this transformation, we compare \(Z\) values against the unified standard distribution values, allowing the use of pre-calculated tables to find probabilities quickly.

Probability calculation is a critical aspect of determining likelihoods in uncertain scenarios. In the given problem, we're tasked with finding probabilities related to the position and speeds of airplanes.Firstly, for part (a), we need to calculate the probability that the second plane does not overtake the first plane. Mathematically, this means finding the probability \( P(Z > -5) \). By converting to the standard normal distribution using the transformation, we find \( P(Z' > -1.77) \), which corresponds to probabilities given by standard normal tables.

Secondly, part (b) asks for the probability that the planes are separated by at most 10 km. This involves calculating probabilities within a certain range: \( -5 \leq Z \leq 0 \). This requires calculating \( P(-1.77 \leq Z' \leq -1.41) \) after converting to a standard normal variable. Here, \( P \) values are obtained by subtracting cumulative probabilities given in standard tables.

This process showcases how statistical tools convert real-world problems into mathematical expressions that can be computed to assist in decision-making.