Unit 9 of 13
Continuous Probability and Its Mathematical Tools
- 6 lessons
- Videos 58 to 63
Continuous quantities spread probability over intervals. This unit introduces the integrals and derivatives that densities need, then builds densities, CDFs, mixed distributions and moments.
Probabilities become areas under a density, found from Riemann sums and definite integrals of simple powers, and where a density is continuous, the derivative of the CDF gives back the density. A density is nonnegative with total area one, and each single value then has probability zero. Mixed distributions combine a point mass with a continuous part, so the endpoints of an event can matter again. Expected value and variance become integrals, with checks that they exist, and the continuous uniform distribution brings these tools together with quantiles, mean and variance.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Integrals and Derivatives for Probability
Integrals and Probability as Area
How a probability density spreads probability along a scale, how Riemann sums and the definite integral measure the area under it, and how the power rule gives accumulated and interval probabilities.
Video 58
Derivatives and the Density-CDF Connection
How difference quotients and derivatives measure local change, how the fundamental theorem of calculus ties integrals to antiderivatives, and when differentiating a CDF gives back its density.
Video 59
Densities and Mixed Distributions
Probability Densities and CDFs
What makes a function a probability density, why density heights carry units and can exceed 1, how to write the complete CDF of an absolutely continuous variable, and why each single value has probability zero.
Video 60
Point Probabilities and Mixed Distributions
How a point mass puts positive probability on one value, how a mixed distribution adds a spread-out part, how CDF jumps and left limits give point probabilities, and what probability zero and almost surely mean.
Video 61
Continuous Moments and Uniform Distributions
Continuous Expectation and Variance
How the mean and variance of a continuous variable become integrals against its density, how the law of the unconscious statistician and mixed distributions fit in, and when a moment is infinite or fails to exist.
Video 62
Uniform Distributions
How a continuous uniform distribution spreads probability evenly over an interval, how to find its interval probabilities, quantiles, percentiles, median, mean and variance, and how conditioning and a change of units affect it.
Video 63