Question

If you sample one number from a standard normal distribution, what is the probability it will be \(0.5 ?\)

Short answer

The probability is 0.

Step-by-step solution

Understanding the Standard Normal Distribution

A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is a continuous probability distribution.

Recognizing the Issue with Discrete Values

In a continuous distribution, the probability of getting an exact single value (like 0.5) is technically zero. This is because there are an infinite number of possible values that can be assumed by a continuous variable, making the probability of any single value infinitesimally small.

Defining Probability for Continuous Distributions

When working with continuous distributions, probabilities are defined over intervals rather than exact values. This means we calculate the probability of a range of values, not the probability of one singular value.

Conclusion for Exact Values

Since the single value 0.5 is part of the continuum of real numbers in a standard normal distribution, the exact probability that the value of a standard normal random variable is exactly 0.5 is 0.

Key concepts

Mean of Zero

In a standard normal distribution, the concept of a "mean of zero" is central to its definition. The mean is a measure of central tendency and represents the average value by balancing the data set. For a standard normal distribution, the mean is exactly zero. This implies that the distribution is symmetric around the midpoint of zero.

Therefore, the mean represents where the highest peak (or the center) of the bell-shaped curve lies. Since the mean is zero, the majority of the data values tend to cluster around this central point, and the probabilities decrease as you move away from it. This understanding of mean as the central value helps you grasp why the distribution is so widely balanced.

Why zero? In cases of standard normal distribution, zero makes calculations elegant and simplifies interpretation. There is no prior or average measure influencing the probability of certain outcomes.

Standard Deviation of One

The standard deviation is another core concept when discussing the standard normal distribution. It measures how spread out the numbers in the data set are. In a standard normal distribution, the standard deviation is consistently set at one.

This property means that the spread of the data values from the mean (which is zero) is consistently scaled. Essentially, a standard deviation of one unit indicates that most of the values lie close to the mean, forming a well-defined shape that doesn't change from one standard deviation unit to the next.

• Why use one? Setting the standard deviation to one provides a standard reference for comparing different data sets and facilitates statistical computations.
• Impact: The values further from the mean (1 unit of standard deviation away) appear less frequently.

In practice, these properties make the standard normal distribution a reference point in statistics, helping to standardize various distributions and enabling us to calculate probabilities with standardized tools.

Continuous Probability Distribution

The nature of a continuous probability distribution, like the standard normal distribution, is a bit different from its discrete counterparts. In a continuous distribution, the possibilities aren't limited to countable outcomes. Instead, there are infinite outcomes between any given two numbers within the range.

So, when considering probabilities in such a distribution, we must think in terms of intervals, not exact values. Let's illustrate this further:

• An exact value in a continuous spectrum, like attempting to find the probability of precisely 0.5, is technically 0. The tiny width of "exact" values makes their occurrence nearly impossible because there are infinite potential outcomes.
• Instead: We calculate probabilities over ranges. For example, the chance that a value falls between 0.4 and 0.6 is meaningful, unlike exact point probabilities.

This fundamental understanding aids in interpreting a continuous distribution like the standard normal distribution. It guides you to think about the probability density function, which describes the likelihood of a variable taking a specific range of values rather than one singular point.