To find the probability between two numbers in a continuous probability distribution, calculate the integral of the probability density function over the specified range.

Finding the probability between two numbers in a continuous probability distribution involves integrating the probability density function (PDF) over the specified range. Here’s a detailed explanation:

Continuous Probability Distribution

In a continuous probability distribution, the probability of a specific point is usually zero due to the infinite number of possible values. Instead, probabilities are associated with intervals or ranges.

Probability Density Function (PDF)

The PDF represents the likelihood of a random variable taking on a specific value. The area under the curve of the PDF within a given interval corresponds to the probability of the variable falling within that interval.

Steps to Find Probability Between Two Numbers

1. Identify the Probability Density Function (PDF):
   • Understand the distribution and determine the mathematical expression for the PDF.
2. Define the Interval:
   • Specify the range or interval between the two numbers for which you want to find the probability.
3. Set Up the Integral:
   • Write the integral to calculate the probability. For a continuous distribution, this involves integrating the PDF over the specified interval.
     P (a ≤ X ≤ b) = ∫_a​^b f(x) dx
   • Here, f(x) is the PDF, and [a, b] is the interval of interest.
4. Perform the Integration:
   • Evaluate the integral using the antiderivative (cumulative distribution function – CDF) of the PDF.
     P(a ≤ X ≤ b) = F(b)−F(a)
   • F(x) is the CDF, representing the accumulated probability up to the value x.

Example: Let’s say you have a standard normal distribution with a PDF f(x) = (1 / √2π) * e^{(-x^2 )/ 2}​. To find the probability between -1 and 1, you would set up and evaluate the integral:

P (-1 ≤ X ≤ 1) = ∫_-1¹ (1 / √2π) * e^{(-x^2 )/ 2} dx

Note:

• Ensure that the PDF is properly normalized (integrates to 1 over the entire range) for accurate probabilities.
• Utilize mathematical software or calculators for complex integrations.

By following these steps and understanding the underlying distribution, you can find the probability associated with a specified interval in a continuous probability distribution.