Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept.

1. Probability & Statistics for Engineers & Scientists, byWalpole, Myers, Myers & Ye ~Chapter 4 Notes Class notes for ISE 201 San Jose State University Industrial & Systems Engineering Dept. Steve Kennedy

2. Mean of a Set of Observations • Suppose an experiment involves tossing 2 coins. The result is either 0, 1, or 2 heads. Suppose the experiment is repeated 15 times, and suppose that 0 heads is observed 3 times, 1 head 8 times, and 2 heads 4 times. • What is the average number of heads flipped? • x bar = (0+0+0+1+1+1+1+1+1+1+1+2+2+2+2) / 15 = ((0)(3) + (1)*(8) + (2)*(4)) / 15 = 1.07 • This could also be written as a weighted average, • x bar = (0)(3/15) + (1)(8/15) + (2)(4/15) = 1.07where 3/15, 8/15, etc. are the fraction of times the given number of heads came up. • The average is also called the mean.

3. Mean of a Random Variable • A similar technique, taking the probability of an outcome times the value of the random variable for that outcome, is used to calculate the mean of a random variable. • The mean or expected value of a random variable X with probability distribution f (x), is = E (X) = x x f(x) if discrete, or  = E (X) = x x f(x) dx if continuous

4. Mean of a Random Variable Depending on X • If X is a random variable with distribution f(x). The mean g(X)of the random variable g(X) is g(X) = E [g(X)] = x g(x) f(x) if discrete, or g(X) = E [g(X)] = x g(x) f(x) dx if continuous

5. Expected Value for a Joint Distribution • If X and Y are random variables with joint probability distribution f (x,y). The mean or expected value g(X,Y) of the random variable g (X,Y) isg(X,Y) = E [g(X,Y)] = x y g(x,y) f(x,y) if discrete, org(X,Y) = E [g(X,Y)] = x y g(x,y) f(x,y) dy dx if continuous • Note that the mean of a distribution is a single value, so it doesn't make sense to talk of the mean the distribution f (x,y).

6. Variance • What was the variance of a set of observations? • The variance2 of a random variable X with distribution f(x) is 2 = E [(X - )2] = x (x - )2 f(x) if discrete, or 2 = E [(X - )2] = x (x - )2 f(x) dx if continuous • An equivalent and easier computational formula, also easy to remember, is 2 = E [X2] - E [X]2 = E [X2] - 2 • "The expected value of X2 - the expected value of X...squared." • Derivation from the previous formula is simple.

7. Variance of a Sample • There's also a somewhat similar, better computational formula for s2. • What is s2? • What was the original formula for the variance of a sample? • The formula is

8. Covariance • If X and Y are random variables with joint probability distribution f (x,y), the covariance, XY , of X and Y is defined asXY = E [(X - X)(Y - Y)] • The better computational formula for covariance isXY = E (XY) - X Y • Note that although the standard deviation  can't be negative, the covariance XY can be negative. • Covariance will be useful later when looking at the linear relationship between two random variables.

9. Correlation Coefficient • If X and Y are random variables with covariance XY and standard deviations X and Y respectively, the correlation coefficient XY is defined as XY = XY / ( X Y )Correlation coefficient notes: • What are the units of XY ? • What is the possible range of XY ? • What is the meaning of the correlation coefficient? • If XY = 1 or -1, then there is an exact linear relationship between Y and X (i.e., Y = a + bX). If XY = 1, then b > 0, and if XY = -1, then b < 0. • Can show this by calculating the covariance of X and a + bX, which simplifies to b / b2 = 1.

10. Linear Combinations of Random Variables • If a and b are constants, E (aX + b) = a E(X) + b • Also holds if a = 0 or b = 0. • If we add two functions, E [g(X)  h(X)] = E [g(X)]  E [h(X)] • Also true for functions of two or more random variables. • That is, E [g(X,Y)  h(X,Y)] = E [g(X,Y)]  E [h(X,Y)]

11. Functions of Two or More Random Variables • The expected value of the sum of two random variables is equal to the sum of the expected values. E (X  Y) = E(X)  E(Y) • The expected value of the product of two independent random variables is equal to the product of the expected values. E (X Y) = E(X) E(Y)

12. Variance Relationships • For a random variable X with variance 22aX + b = a2 X2 • So adding a constant does what? • And multiplying by a constant does what? • For two random variables X and Y,2aX + bY = a2 X2 + b2 Y2 + 2abXY • What if X and Y are independent? • XY = 0. Note that the correlation coefficient is also 0.

13. Chebyshev's Inequality • The probability that any random variable X will assume a value within k standard deviations of the mean is at least 1 - 1/k2. That is P ( - k < X <  + k)  1 - 1/k2 • This theorem is both very general and very weak. • Very general, since it holds for any probability distribution. • Very weak for the same reason, because it is a worst-case limit that holds for any distribution. • If we know the distribution, we can get a better limit than this (how?), so this is only used when the distribution is unknown. • Care must be taken, however, not to assume an underlying distribution when the distribution is really unknown.