By Klotz J.H.

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7). 2 . 8). 4. 25 for these sorted Walsh sums. from which the median is M The sample mean is . 521 . 9 K D= i=1 . 4697 . 12) we obtain . 74985 32 CHAPTER 1. 2 2 S {1} = 0 . 48571 . 07143 S 2 {2} = 3−1 7−1 . 78667 . 07143 + 15×14 . 9520 . 03510 S 2 {4} = 15−1 25−1 . 16129 . 32445 . 52121 . 14) Properties If the data values are transformed by a linear transformation Xi → aXi + b then these measures of the center transform in the same way ˜ → aX ˜ + b, Tr → aTr + b, Wr → aWr + b, M ˜ → aM ˜ + b, X ¯ → aX ¯ + b.

We consider some important such random variables. 1. 1 63 Binomial Random Variables Consider the sample space of tossing a bent coin n times with outcome probabilites determined by P (H) = p and P (T ) = 1 − p = q where 0 < p < 1. Then P (HHH · · · H T T T · · · T ) = px q n−x x n−x and similarly for all outcomes with x H’s and n − x T ’s. Then if X is the number of heads observed we have P (X = x) = n x px q n−x for x = 0, 1, 2, . . , n n such outcomes with probability px q n−x . This distrix bution is called the binomial distribution.

Let B be the event that all 5 cards are red {♥, ♦}. Then P (A) = 4× 13 5 52 5 = 5148 . 4. 2: Changing Sample Space for Conditional Probability. and P (A|B) = 2× 52 13 / 5 5 26 52 / 5 5 = 2× 13 5 26 5 = 2574 . 039130 . 65780 The probability of A knowing B has occurred increases, in this case, since mixtures with black cards have been ruled out. An important theorem for conditional probabilities is Bayes Theorem. Let the sample space be partitioned into disjoint subsets K S= k=1 Bk where Bj ∩ Bk = φ for j = k .