Statistics (university) · Part 04
Continuous distributions and the normal curve
Model measurements with density curves, the uniform, the exponential and the bell curve.
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- Probability density functionA probability density function (PDF) is a curve for a continuous variable. The area under the curve between two values is the probability of landing between them. The total area is 1. The chance of any exact single value is 0.
- Uniform distributionA uniform distribution on the range a to b makes every stretch of equal width equally likely. Its mean is (a + b) / 2 and its variance is (b - a) squared / 12.
- Exponential distributionThe exponential distribution models the waiting time until the next event when events occur at rate lambda. P(X > t) = e^(-lambda t). The mean is 1/lambda. It is the waiting-time partner of the Poisson.
- Normal distributionThe normal distribution is the symmetric bell curve, set by its mean mu and standard deviation sigma. About 68 percent of values lie within 1 sigma of the mean, 95 percent within 2 sigma and 99.7 percent within 3 sigma.
- Standard normal and z-scoresThe standard normal has mean 0 and SD 1. Any normal value X is converted by Z = (X - mu) / sigma. Then a z table or calculator gives the area to the left, P(Z <= z).
- Normal probabilities and percentilesTo find a probability, convert to z and read the area. To find a percentile, work backwards: find the z for the area, then compute X = mu + z x sigma.
- Normal approximation to the binomialA binomial with large n can be approximated by a normal with mean np and SD square root of np(1-p). It works well when np and n(1-p) are both at least 10. Add or subtract 0.5 (continuity correction) to bridge whole numbers and a smooth curve.
- Sums of normal variablesIf X and Y are independent normals, X + Y is also normal. Means add, and variances add: mean = mu1 + mu2, variance = sigma1 squared + sigma2 squared. Standard deviations do not add.