2 interactive labs · AS & A Level Mathematics 9709
Probability Density and Area
6.3 Continuous random variables · Find k so the total area under f(x) is 1, shade P(a < X < c), then find the median or a percentile by making the area to its left equal to a given value.
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About this lab
Find k so the total area under f(x) is 1, shade P(a < X < c), then find the median or a percentile by making the area to its left equal to a given value.
You will be able to
Determine the constant k in a probability density function by setting ∫ f(x) dx = 1 over the stated interval, including an infinite interval such as k/x⁴ for x ≥ 1.
Calculate P(a < X < c) by integrating the density between the bounds, and explain why P(X = a) = 0 for a continuous random variable.
Find the median or another percentile by solving 'area to the left = ½' (or the given area), and reject a root that lies outside the interval.
Mean and Variance from a Density
6.3 Continuous random variables · Sweep the upper limit to build E(X) and E(X²) as weighted areas, then find Var(X) = E(X²) − [E(X)]² and compare the mean, median and mode.
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About this lab
Sweep the upper limit to build E(X) and E(X²) as weighted areas, then find Var(X) = E(X²) − [E(X)]² and compare the mean, median and mode.
You will be able to
Calculate E(X) = ∫ x f(x) dx and E(X²) = ∫ x² f(x) dx for a density on one interval, including an infinite interval such as 3/x⁴ for x ≥ 1.
Calculate Var(X) = E(X²) − [E(X)]² from the two integrals, and explain why E(X²) − E(X) is wrong.
Predict, from the shape of the density, whether the mean is below, equal to or above the median, and compare the mean, median and mode of a skewed distribution.
