Continuous Random Variables: PDF, CDF, and Percentiles
Master continuous distributions, density functions, and percentile calculations for Exam P.
Probability Density Function
A continuous random variable X has a probability density function (PDF) f(x) satisfying f(x) >= 0 and the integral of f(x) over all real numbers equals 1. Unlike a PMF, f(x) is not a probability itself. Instead, P(a < X < b) = integral of f(x) from a to b. A consequence is that P(X = a) = 0 for any single point.
Common Exam P tasks include finding a normalizing constant. If f(x) = c * x^2 for 0 < x < 3, then integrating c * x^2 from 0 to 3 gives c * 9 = 1, so c = 1/9.
CDF and Percentiles
The CDF is F(x) = P(X <= x) = integral of f(t) from negative infinity to x. F is non-decreasing with F(negative infinity) = 0 and F(positive infinity) = 1. To find the PDF from the CDF, differentiate: f(x) = F'(x).
The p-th percentile (or quantile) x_p satisfies F(x_p) = p. For example, the median is the 50th percentile where F(x) = 0.5. Exam P problems often ask you to solve F(x) = 0.5 for x. If F(x) = x^3/27 on [0,3], then the median satisfies x^3/27 = 0.5, giving x = (13.5)^(1/3) = 2.381.
Expected Value and Variance for Continuous Variables
E[X] = integral of x * f(x) dx and Var(X) = E[X^2] - (E[X])^2, analogous to the discrete case. Integration by parts is a common technique on Exam P. For survival-type distributions on [0, infinity), a useful shortcut is E[X] = integral from 0 to infinity of (1 - F(x)) dx, often called the survival function method for computing the mean.