Quick review of some key ideas

1. Quick review of some key ideas CEE 11 Spring 2002 Dr. Amelia Regan These notes draw liberally from the class text, Probability and Statistics for Engineering and the Sciences by Jay L. Devore, Duxbury 1995 (4th edition)

2. mean • The mean of a sample or data set is simply the arithmetic average of the values in the set, obtained by summing the values and dividing by the number of values. • The mean of the sample of weights is 144.63 pounds

3. mean of a frequency distribution • When we summarize a data set in a frequency distribution, we are approximating the data set by "rounding" each value in a given class to the class mark. • The mean of the weight data obtained in this way is 146.67

4. shortcut formula for the variance • Its sometimes more convenient to use the following formula for the variance

5. properties of S2 • Let x1, x2, x,...,xn be a sample and c be any nonzero constant. • If y1 = x1 + c, y2 = x2 + c,...,yn = xn + c, then S2y = S2x • If y1 = cx1, y2 = cx2,...,yn = cxn, then S2y = c2S2x, Sy = |c|S2x • In other words -- if we add a constant to a sample we do not increase the variance -- if we multiply by a constant we increase the variance by the square of the constant

6. related properties of the sample mean • Let x1, x2, x,...,xn be a sample and c be any nonzero constant. • If y1 = x1 + c, y2 = x2 + c,...,yn = xn + c then • If y1 = cx1, y2 = cx2,...,yn = cxn, then • In other words if we add or multiply the sample by a constant we add or multiply the mean by the same constant

7. Class exercise (new) • Without using your calculators, calculate the mean and variance of the following data • Xi | 33 27 18 36 21 • ---------------------------------------------- • fi | 4 3 2 5 1 • Hint, shift the observations “to the left” by subtracting a constant and then divide by another constant

8. Class exercise (new) • Without using your calculators, calculate the mean and variance of the following data • Step 1 yi | 11 9 6 12 7 • ---------------------------------------------- • Step 2 yi | 2 0 -3 3 -2 • yi | 2 0 -3 3 -2 • ---------------------------------------------- • fi | 4 3 2 5 1 • Divide by 3, then subtract 9

9. yi | 2 0 -3 3 -2 • -------------------------------------------------------- • (yi)2 | 4 0 9 9 4 • -------------------------------------------------------- • fi | 4 3 2 5 1

11. conditional probability • For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by • The multiplication rule for P(A intersection B) follows directly

12. independence • Remember that in general • However, if A and B are independent then • In fact, A and B are independent if and only if the above is true

13. Counting Techniques • When the various outcomes of an experiment are equally likely then the task of computing probabilities reduces to counting. In particular, if N is the number of outcomes in the sample space and N(A) is the number of outcomes contained in an event A, then

15. Permutations • Any ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects. The number of permutations of size k that can be constructed from the n objects is denoted by Pk,n • The number of permutations of size k that can be constructed from n objects is equal to n(n-1)(n-2)…(n-k+1)

16. combinations • Given a set of n distinct objects, any unordered subset of size k of the objects is called a combination. The number of combinations of size k that can be formed from n distinct objects is denoted by or sometimes by Ck,n