Which function is used to describe the distribution of probabilities for continuous random variables?

Which function is used to describe the distribution of probabilities for continuous random variables?

• A. Probability Mass Function
• B. Probability Density Function
• C. Cumulative Distribution Function
• D. Moment Generating Function

Answer: (B)

For continuous random variables, the distribution of probabilities is described by the probability density function. The key idea is that probabilities aren’t assigned to individual points (since the chance of hitting an exact value is zero); instead, the density tells you how the probability is spread over values. The probability that X falls in an interval [a, b] is found by integrating the density over that interval. The density must be nonnegative and, when integrated over the entire support, must equal 1. A cumulative distribution function, while related, gives P(X ≤ x) and can describe the distribution in a different form, but it’s not the direct density that assigns probability to tiny intervals. The moment generating function encodes moments like the mean and variance, not the distribution itself. The probability mass function applies to discrete variables, where probabilities are assigned to individual values. So the function that directly describes how probability is distributed across values for a continuous variable is the probability density function.

For continuous random variables, the distribution of probabilities is described by the probability density function. The key idea is that probabilities aren’t assigned to individual points (since the chance of hitting an exact value is zero); instead, the density tells you how the probability is spread over values. The probability that X falls in an interval [a, b] is found by integrating the density over that interval. The density must be nonnegative and, when integrated over the entire support, must equal 1.

A cumulative distribution function, while related, gives P(X ≤ x) and can describe the distribution in a different form, but it’s not the direct density that assigns probability to tiny intervals. The moment generating function encodes moments like the mean and variance, not the distribution itself. The probability mass function applies to discrete variables, where probabilities are assigned to individual values.

So the function that directly describes how probability is distributed across values for a continuous variable is the probability density function.