Mathematical Expectation

Mean of a Random Variable

A coin is biased such that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice

Variance and Covariance of Random Variables

• The mean, or expected value, of a random variable is of special importance

• It describes where the probability distribution is centered.

• the mean does not give an adequate description of the shape of the

distribution.

• also need to characterize the variability in the distribution.

• the histograms of two discrete probability distributions

• have the same mean, μ = 2,

• but differ considerably in variability,

• the dispersion of their observations about the mean.

• variance of the random variable X

•or the variance of the probability distribution of X

•denoted by Var(X) or the symbol σ2

All of them have mean µ = 3.

All of them have mean µ = 3.

Covariance:• measure of how much two random variables vary together.• height and weight of giraffes have positive covariance• because when one is big the other tends also to be big.

Correlation:• units of covariance are ‘units of X times units of Y ’. • hard to compare covariances: • if we change scales then the covariance changes as well. • Correlation is a way to remove the scale from the covariance.

Means and Variances of Linear Combinations ofRandom Variables

develop some useful properties that will simplify the calculations of means and variances of random variables

properties will permit us to deal with expectations in terms of other parameters that are either known or easily computed.

• If X represents the daily production of some item from machine A

• and Y the daily production of the same kind of item from machine B,

• then X + Y represents the total number of items produced daily by both machines

• considering the experiment of tossing a green die and a red die. • random variable X represent the outcome on the green die • random variable Y represent the outcome on the red die. • Then XY represents the product of the numbers that occur • In the long run, the average of the products of the numbers• is equal to the product of the average number on the green die and the

average number on the red die

• variance is unchanged if a constant is added to or subtracted from a

random variable.

• The addition or subtraction of a constant simply shifts the values of X

• to the right or to the left but does not change their variability.

corollaryn. A proposition that follows with little or no proof required from one already proven.n. A deduction or an inference.n. A natural consequence or effect; a result.

theoremn. An idea that has been demonstrated as true or is assumed to be so demonstrable.n. Mathematics A proposition that has been or is to be proved on the basis of explicit assumptions

• if a random variable is multiplied or divided by a constant,

• then Corollaries state that the variance is multiplied or

divided

• by the square of the constant.