MoreLikelihood

Review

• Random number generators (rand and randn) produced differently distributed pseudo-random numbers.

• Setting the random seed allows you to reproduce the same random numbers.

• Remember the help and type commands, these are useful for learning MATLAB. Here's another useful one:

lookfor random

• Statistics: mean, median, mode, std, min, max. These MATLAB commands are functions of data, but they have theoretical counterparts which depend upon the PMF or PDF.

• For discrete random variables: Probability Mass Function which is the function of the possible values of random variable equal to the fraction of throws that land on the value.

• For continuous random variables: Probability Density Function which is the function of the possible values of the random variable equal to the fraction of throws that land in a bin divided by width of bin, or actually the limit of this fraction as the bin width (bin containing the value) converges to 0.

• Not review but extension: The axiomatic theory of probability always has in mind a set of outcomes of the experiment at hand. A random variable is a "real valued" function of the possible outcomes of the experiment. The possible values of the random variable might be the same as the possible outcomes, or there might be more outcomes. Example: Say the experiment is to throw a dart at a dartboard and the random variable is the number of points awarded. The values of the random variable are the integers between 0 and 100. The possible outcomes of the experiment could be the number of points awarded or it could be the (Cartesian or polar) coordinates of the dart on the board. In the latter case, infinitely many outcomes yeild the same value of the random variable.

• Following up the last 3 points, the PMF and PDF are always functions mapping all real numbers to probabilites (between 1 and 0) or probability densities (between 0 and infinity).

• Functions of random variables lead to new random variables.

• Sometimes it is the case that you observe something random (a person's performance throwing darts) and you want a statistical model. You specify a PDF (for the dart's coordinates) but since you are just guessing to leave free some parameters. You have an assumed probability density (of the dart's coordinates) that depends upon the dart's coordinates and parameters. If you fix the parameters, you get a PDF, a function of just the dart's coordinates (possible outcomes/values of random variable). If you fix the cooridinates, you get a likelihood function, a function of the parameters. But is not a PDF: the parameters are not random they are just unknown.