Functions of random variables Sometimes what we can measure is not what we are interested in! Example: mass of binary-star system: We want M but can only measure V and P. Must conserve probability: Y X f(X) f(Y)

Non-linear transformations e.g.Flux distributions vs. wavelength, frequency: Fluxes and magnitudes: –Gaussian distribution: X ~ G(X 0,  2 ) –Nonlinear transformation induces a bias: –PROBLEM: evaluate a,  (M) in terms of X 0, . X M=-2.5 log X f(M) f(X)

Nonlinear transformations bias the mean To find, use Taylor expansion around X= : Hence X Y f(Y) f(X) Y=g(X) 0 This is the bias.

Variance of a transformed variable Get variance of Y from first principles: X Y f(Y) f(X) Y=g(X)

What is a statistic? Anything you measure or compute from the data. Any function of the data. Because the data “jiggle”, every satistic also “jiggles”. Example: the mean value of a sample of N data points is a statistic: It has a definite value for a particular dataset, but it also “jiggles” with the ensemble of datasets to trace out its own PDF. NB:

Sample mean and variance - 1 Sample mean: The distribution of sample means has a mean:...and a variance: if the X i are independent

Sample mean and variance - 2 If the X i are all drawn from a single parent distribution with mean and variance  2, then: And:

Other unbiased statistics Sample median (half points above, half below) (X max + X min ) / 2 Any single point X i chosen at random from sequence Weighted average:

Inverse variance weighting is best! Let’s evaluate the variance of the weighted average for some weighting function w i : The variance of the weighted average is minimised when: Let’s verify this -- it’s important!

Choosing the best weighting function To minimise the variance of the weighted average, set:

Using optimal weights Good principles for constructing statistics: –Unbiased -> no systematic error –Minimum variance -> smallest possible statistical error Optimally (inverse-variance) weighted average: Is unbiased, since: And has minimum variance: