For continuous random variables, we use which function?

For continuous random variables, we use which function?

• A. Cumulative distribution function
• B. Moment generating function
• C. Probability density function
• D. Probability mass function

Answer: (C)

For a continuous random variable, probabilities are assigned to ranges of values and are described by a density function. The probability density function (pdf) assigns nonnegative values f(x) with the total area under the curve equal to 1, and probabilities over intervals are found by integrating the density: P(a ≤ X ≤ b) = ∫_a^b f(x) dx. The cumulative distribution function (CDF) is related, since F(x) = P(X ≤ x) = ∫_{-∞}^x f(t) dt, but the fundamental object that describes the distribution in terms of density is the pdf itself. The other options describe different concepts: a probability mass function is for discrete variables and assigns probabilities to specific points; the moment generating function encodes moments like mean and variance but isn’t the density; the CDF is the accumulated probability up to a point, not the density. For example, with a continuous distribution like X ~ Exp(1), the pdf is f(x) = e^{-x} for x ≥ 0, and probabilities are areas under this curve, while the CDF would be F(x) = 1 − e^{-x} for x ≥ 0.

For a continuous random variable, probabilities are assigned to ranges of values and are described by a density function. The probability density function (pdf) assigns nonnegative values f(x) with the total area under the curve equal to 1, and probabilities over intervals are found by integrating the density: P(a ≤ X ≤ b) = ∫a^b f(x) dx. The cumulative distribution function (CDF) is related, since F(x) = P(X ≤ x) = ∫{-∞}^x f(t) dt, but the fundamental object that describes the distribution in terms of density is the pdf itself. The other options describe different concepts: a probability mass function is for discrete variables and assigns probabilities to specific points; the moment generating function encodes moments like mean and variance but isn’t the density; the CDF is the accumulated probability up to a point, not the density. For example, with a continuous distribution like X ~ Exp(1), the pdf is f(x) = e^{-x} for x ≥ 0, and probabilities are areas under this curve, while the CDF would be F(x) = 1 − e^{-x} for x ≥ 0.