C HAPTER 7 Section 7.2 – Means and Variances of Random Variables

M EANS AND V ARIANCES OF R ANDOM V ARIABLES Probability is the mathematical language that describes the long-run regular behavior of random phenomena. The probability distribution of a random variable is an idealized relative frequency distribution.

E XAMPLE T HE T RI -S TATE P ICK 3

T HE M EAN OF A R ANDOM V ARIABLE

M EAN OF A D ISCRETE R ANDOM V ARIABLE

M EAN AND E XPECTED V ALUE

T HE V ARIANCE OF R ANDOM V ARIABLE

Definition: Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 … Probability: p 1 p 2 p 3 … and that µ X is the mean of X. The variance of X is The standard deviation σ x of X is the square root of the variance. T HE V ARIANCE OF R ANDOM V ARIABLE

E XAMPLE S ELLING A IRCRAFT P ARTS Gain Communications sells aircraft communications units to both the military and the civilian markets. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The military division estimates its sales as follows: Units sold: ,000 Probability: Calculate the mean and variance of X See p.411 to check your answers

S TATISTICAL E STIMATION AND THE L AW OF L ARGE N UMBERS

L AW OF L ARGE N UMBERS

T HE “L AW OF S MALL N UMBERS ” Both the rules of probability and the law of large numbers describe the regular behavior of chance phenomena in the long run. Psychologists have discovered most people believe in an incorrect “law of small numbers” That is, we expect even short sequences of random events to show the kind of average behavior that in fact appears only in the long run.

H OW L ARGE IS A L ARGE N UMBER ? The law of large numbers says that the actual mean outcome of many trials gets close to the distribution mean μ as more trials are made. It doesn’t say how many trials are needed to guarantee a mean outcome close to μ.

Homework: p #’s 24, 29, & 32 Work on ch.6 & 7 extra credits