Calculating Probabilities of a Normal Distribution

A random variable x is said to have a normal distribution with a mean (μ) and standard deviation (σ), denoted as N(μ, σ2). In this article, we will explore how to calculate the probability of a random variable falling within a specific range in a normal distribution. Specifically, we will use a normal distribution with a mean of 5.6 and standard deviation of 1.4.

Understanding the Problem

We are given a random variable x with a normal distribution N(5.6, 1.42). The task is to calculate the following probabilities:

P(5 x 6) P(x 7) P(x 6.4)

These calculations can be done using the properties of the normal distribution and the concept of Z-scores, which convert values to the standard normal distribution N(0, 1).

Using Z-Score to Convert to Standard Normal Distribution

The Z-score allows us to standardize values from a normal distribution to a standard normal distribution, which has a mean of 0 and a standard deviation of 1. The formula to convert a value x to its Z-score is:

Z (x - μ) / σ

Calculating P(5 x 6)

To find P(5 x 6), we convert the values to Z-scores:

Z1 (5 - 5.6) / 1.4 ≈ -0.4286 Z2 (6 - 5.6) / 1.4 ≈ 0.2857

Using the standard normal distribution table, we find:

P(Z 0.2857) ≈ 0.6141 P(Z -0.4286) ≈ 0.3365

Hence, P(5 x 6) P(0 Z 0.2857) 0.6141 - 0.3365 ≈ 0.2776

Calculating P(x 7)

For the probability P(x 7), we convert the value:

Z (7 - 5.6) / 1.4 ≈ 0.99998

From the standard normal distribution table, we find:

P(Z 0.99998) ≈ 0.8413

Hence, P(x 7) 1 - P(Z 0.99998) ≈ 1 - 0.8413 0.1587

Calculating P(x 6.4)

To find P(x 6.4), we convert the value:

Z (6.4 - 5.6) / 1.4 ≈ 0.5714

From the standard normal distribution table, we find:

P(Z 0.5714) ≈ 0.7157

Hence, P(x 6.4) ≈ 0.7157

In summary, we have calculated the probabilities for the given normal distribution using Z-scores and standard normal distribution tables. These calculations are fundamental in understanding the behavior of normally distributed data and are widely used in various fields such as statistics, engineering, and data science.