Exploring the Normal Distribution and Its Properties
In the realm of probability theory and statistics, the normal distribution, also known as the Gaussian distribution, holds immense importance. It is widely used to model real-world phenomena due to its desirable properties. In this article, we will delve into how a random variable with a normal distribution can be linked to the standard normal distribution, explore the role of the cumulative distribution function (CDF), and discover how to determine the standard deviation from given probabilities.
Understanding the Normal Distribution
A random variable with the normal distribution can be described by its mean μ and standard deviation σ. However, when dealing with standardized forms, it is common to work with the standard normal distribution, denoted as U, which has a mean of 0 and a standard deviation of 1.
Probability Statements and Equivalence
Consider a random variable X that follows a normal distribution with mean μ and standard deviation σ. The cumulative distribution function (CDF) of X is denoted as Φ(x) Pr(X ≤ x). We can express various probability statements using the properties of the normal distribution and the CDF of the standard normal distribution.
Example of Equivalence in Probabilities
Given a random variable X with the normal distribution, the following statements are equivalent:
Pr(X1 ≤ X ≤ 3) 0.75
This means that the probability of X lying between 1 and 3 is 0.75.
Pr(?1 ≤ X ? 2 ≤ 1) 0.75
By translating the above statement by 2 units to the right, we shift the interval from (1, 3) to (3, 5). This implies that the probability of X being within 1 unit of 2 is 0.75.
Pr(?1/σ ≤ (X - 2)/σ ≤ 1/σ) 0.75
Normalizing the interval by dividing by σ results in a statement involving the standard normal distribution U. Here, we consider the interval of U between -1/σ and 1/σ.
Pr(?1/σ ≤ U ≤ 1/σ) 0.75
Using the CDF of the standard normal distribution, we can write:
Φ(1/σ) - Φ(?1/σ) 0.75
where Φ is the CDF of the standard normal distribution.
Standard Normal Distribution Symmetry
One of the key properties of the standard normal distribution is its symmetry about the mean, which is 0. Specifically, the probability of a value being less than or equal to -u is the same as the probability of a value being greater than or equal to u. This symmetry is expressed as:
Pr(U ≤ ?u) Pr(U ≥ u)
Consequently, the CDF of the standard normal distribution at -1/σ is:
Φ(?1/σ) 1 - Φ(1/σ)
Using this information, we can find the value of the CDF at 1/σ.
Evaluating CDF for Specific Probabilities
Given the above results, we can determine the value of Φ(1/σ). If we know the probability corresponding to this interval, we can use a standard normal distribution table (also known as the Z-table) to find the value of 1/σ.
From the statement, we have:
Φ(1/σ) - Φ(?1/σ) 0.75
Using the symmetry property:
Φ(1/σ) - (1 - Φ(1/σ)) 0.75
2Φ(1/σ) - 1 0.75
2Φ(1/σ) 1.75
Φ(1/σ) 0.875
With the value of Φ(1/σ) known, we can now find 1/σ using the Z-table, and subsequently determine the standard deviation σ.
Conclusion: The exploration of the normal distribution and its properties, particularly through the standard normal distribution, provides a powerful tool for statistical inference. By understanding and applying these properties, one can effectively determine key parameters like the standard deviation, enhancing the accuracy and reliability of probabilistic models.