Continuous Random Variables and Probability Density Functions Explained

For continuous random variables: (1) E[X] = ∫ₐᵇ x × f(x)dx, (2) E[X²] = ∫ₐᵇ x² × f(x)dx, (3) Var(X) = E[X²] - (E[X])², (4) σ = √Var(X). For uniform distribution on [0, 4]: E[X] = 2, E[X²] = 16/3, Var(X) = 4/3, σ = √(4/3). Variance must be ≥ 0.

For continuous random variables, the expected value (mean) μ is calculated as the integral from negative infinity to positive infinity of x times f(x) dx. The variance σ² measures the average squared distance from the mean and is calculated as the integral from negative infinity to positive infinity of (x - μ)² times f(x) dx. The standard deviation is the square root of the variance. These formulas are analogous to their discrete counterparts but use integration instead of summation.

For continuous random variables: (1) Expected value (mean) is calculated as the integral of x times the probability density function over the interval, (2) Variance is calculated as the integral of (x minus the expected value) squared times the probability density function, and (3) Standard deviation is the square root of the variance.

For continuous random variables, the expected value (mean) is calculated using integration over the range of the variable: E[X] = ∫x·f(x)dx. The variance is calculated as Var(X) = E[X²] - (E[X])², where E[X²] requires integrating x²·f(x) over the range. Standard deviation is the square root of variance.

The expected value (mean) of a continuous random variable is calculated as E(X) = ∫_a^b x·f(x) dx. The variance is calculated as Var(X) = E(X²) - [E(X)]², where E(X²) = ∫_a^b x²·f(x) dx. The standard deviation is the square root of the variance, measuring the spread of the probability distribution. For example, if f(x) = 1/3 for 0 ≤ x ≤ 3, then E(X) = 1.5 and E(X²) = 4.5, so Var(X) = 4.5 - (1.5)² = 2.25 and standard deviation = √2.25 = 1.5.

This section covers three continuous probability distributions. The uniform distribution describes equally likely outcomes in an interval [a,b], with PDF f(x) = 1/(b-a) and mean (a+b)/2. The exponential distribution models time between Poisson events, with PDF f(x) = λ × e^(-λx) and memoryless property. The normal distribution (Gaussian) is characterized by its bell-shaped curve with PDF f(x) = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²)). Key properties include symmetry about the mean, 68% within ±1 standard deviation, 95% within ±2, and 99.7% within ±3. The standard normal distribution has μ=0 and σ=1, with Z-score transformation Z = (X - μ)/σ.

Continuous probability distributions model variables taking any value within intervals. The uniform distribution on [A, B] has PDF f(x) = 1/(B-A), mean (A+B)/2, and variance (B-A)²/12. The exponential distribution models time between events with PDF f(x) = λ·e^(-λx), mean 1/λ, variance 1/λ²; shifted versions have mean a+B and variance B². The normal (Gaussian) distribution is symmetric and bell-shaped with PDF f(x) = (1/(σ√(2π)))·e^(-(x-μ)²/(2σ²)), mean μ, variance σ². The normal distribution is fundamental because many natural phenomena follow it, and the central limit theorem states sums of independent random variables tend toward normality. For mean μ=0, the probability X > 0 is exactly 1/2 due to symmetry.

Continuous probability distributions describe random variables that take values in continuous intervals of real numbers, with probabilities calculated through integration of a probability density function (PDF). The PDF must be non-negative and integrate to 1 over all real numbers. Three fundamental continuous distributions are: (1) Uniform distribution on [a,b] with PDF f(x) = 1/(b-a), mean (a+b)/2, and variance (b-a)²/12; (2) Exponential distribution with PDF f(x) = λe^(-λx) for x ≥ 0, mean 1/λ, and variance 1/λ², commonly modeling waiting times or lifetimes; (3) Normal distribution with PDF f(x) = (1/(σ√(2π)))e^(-(x-μ)²/(2σ²)), mean μ, and variance σ², which is symmetric about μ and completely characterized by its mean and variance. For any normal variable X ~ N(μ, σ²), probabilities can be calculated by standardizing to the standard normal distribution Z = (X-μ)/σ, using the cumulative distribution function Φ.

This section introduces three fundamental continuous probability distributions. The uniform distribution has constant density f(x) = 1/(b-a) on [a,b], modeling equally likely outcomes like waiting times. The exponential distribution with density f(x) = λe^(-λx) for x≥0 models waiting times between events. The normal distribution, characterized by mean μ and standard deviation σ, has density f(x) = (1/(σ√(2π)))e^(-(x-μ)²/(2σ²)). The standard normal case (μ=0, σ=1) is symmetric about zero. The Central Limit Theorem explains why the normal distribution is fundamental: sums of independent variables tend toward normality.

This comprehensive section covers three fundamental continuous probability distributions. The Uniform Distribution describes variables uniformly distributed over an interval [a, b], with probability density f(x) = 1/(b-a) within the interval. Probabilities are calculated by integrating the density function. The Exponential Distribution models time between events in a Poisson process, with f(x) = (1/μ)e^(-x/μ) for x ≥ 0, where μ is the mean time between events. The Normal Distribution (Gaussian) is the most important continuous distribution, symmetric about mean μ with bell-shaped curve. The probability density function is f(x) = (1/(σ√(2π))) * e^(-(x-μ)²/(2σ²)). Key properties include symmetry about μ, maximum at x = μ, total area equals 1, and approximately 68% of data within one standard deviation. To use standard normal tables, convert any normal distribution using Z = (X - μ)/σ.

Two jointly continuous random variables are described by a joint density function f(x,y) that represents a surface over the x-y plane, with total volume equal to 1. The joint density specifies probabilities of small rectangular regions as approximately f(x,y)×delta², representing probability per unit area. This enables calculation of probabilities for events involving both variables simultaneously, extending univariate concepts to multivariate settings while maintaining the integral-based approach.

For two continuous random variables X and Y, the joint sample space is R² (the Cartesian product of the real numbers with itself). The joint probability density function (pdf) f(x, y) is a function from R×R to R satisfying two properties: (1) f(x, y) ≥ 0 for all x, y, and (2) the double integral of f(x, y) over the entire real plane equals 1. The probability that (X, Y) lies within a region A is equal to the double integral of f(x, y) over A. For a rectangular region [a, b] × [c, d], this probability is computed as the double integral of f(x, y) over that rectangle. The marginal pdf for X is obtained by integrating f(x, y) with respect to y from -∞ to ∞: f_X(x) = ∫_{-∞}^∞ f(x, y) dy. Similarly, f_Y(y) = ∫_{-∞}^∞ f(x, y) dx. The conditional density function of Y given X=x is f_Y|X(y|x) = f(x, y)/f_X(x), and similarly for X given Y=y. The joint pdf satisfies f(x, y) = f_Y|X(y|x) × f_X(x).

The joint probability density function (PDF) for a pair of continuous random variables is defined as a function from which the joint cumulative probability distribution function can be calculated by finding the double integral; unlike discrete variables where marginals are calculated using sums, continuous variables require integrals to compute marginal distributions, and visualizing the joint PDF typically involves two-dimensional contour plots where color represents the probability density values.

Continuous bivariate random variables have joint probability density functions f(x, y) satisfying F(x, y) = ∬_{-∞}^x ∬_{-∞}^y f(s,t) ds dt. The JPDF must be non-negative everywhere with total integral 1. The support is where f(x, y) > 0. For any Borel set B in R², P((X,Y) ∈ B) = ∬_B f(x,y) dx dy. Marginal PDFs are obtained by integrating out the other variable: fₓ(x) = ∫_{-∞}^∞ f(x,y) dy and fᵧ(y) = ∫_{-∞}^∞ f(x,y) dx. Conditional PDFs are fₓ|ᵧ(x|y) = f(x,y)/fᵧ(y) and fᵧ|ₓ(y|x) = f(x,y)/fₓ(x), defined where the marginal PDF is positive.

A joint probability density function (PDF) for two continuous random variables describes the relative likelihood of observing values around a specific point in the joint sample space; it is defined such that the volume under the three-dimensional surface equals 1, and it must satisfy two key conditions: non-negativity (f(x,y) ≥ 0 for all x,y) and the property that the double integral over all possible values equals 1. This concept extends naturally to multivariate cases with n random variables, where the multivariate integral over all components must equal 1 and the function remains non-negative.