Welcome to continuous probability problems. In continuous probability, we work with random variables that can take any value within a range. Unlike discrete probability where we count specific outcomes, continuous probability uses probability density functions, or PDFs, to describe the likelihood of outcomes. The key property is that the total area under the PDF curve equals one, representing the certainty that some outcome will occur.
The uniform distribution is the simplest continuous probability distribution. In a uniform distribution, all values within the interval from a to b are equally likely to occur. The probability density function is constant and equals one divided by b minus a. For example, if we have a uniform distribution from 1 to 4, the height of the rectangle is one-third, ensuring the total area under the curve equals one.
To calculate probabilities in continuous distributions, we find the area under the probability density function between two points. For a uniform distribution, this is straightforward. If we want to find the probability that X is between 1.5 and 3, we calculate the area of the rectangle from 1.5 to 3. Since the height is one-third and the width is 1.5, the probability equals one-half.
The normal distribution is the most important continuous probability distribution. It has a characteristic bell-shaped curve that is symmetric around the mean mu. The formula involves the mathematical constant e and pi. The parameter sigma, called the standard deviation, controls how spread out the distribution is. About 68 percent of values fall within one standard deviation of the mean.
Continuous probability distributions have many practical applications in the real world. Human heights and weights follow normal distributions. Measurement errors in scientific experiments are often normally distributed. Stock prices and financial returns can be modeled using various continuous distributions. In quality control, manufacturers use these concepts to ensure products meet specifications. The 68-95-99.7 rule tells us that most values fall within a few standard deviations of the mean.