Continuous random variables

Learning objectives

Use a probability density function to find probabilities as ∫f(x) dx over an interval, and verify that a proposed function is a valid density by checking that f(x) ≥ 0 throughout and that the integral over the whole range equals 1

Calculate the expectation E(X) = ∫x f(x) dx, the variance Var(X) = E(X²) − [E(X)]², and E(g(X)) = ∫g(x) f(x) dx for a stated function g

Find the cumulative distribution function F(x) = P(X ≤ x) by integrating the density from the lower end of the range, state it piecewise over the whole real line, and recover the density by differentiating F

Use the cumulative distribution function to find the median and the quartiles by solving F(x) = 0.5, 0.25 and 0.75, and identify the mode from the maximum of the density

Find the distribution of a related variable Y = g(X) by first writing P(Y ≤ y) as a probability about X, obtaining the cumulative distribution function of Y, and then differentiating to get its density

Compare the mean, median and mode of a skewed continuous distribution and explain what their relative positions say about the shape of the density curve