Advanced High-School Mathematics

SECTION 6.1 Discrete Random Variables 331 E(X) = r p , Var(X) = r(1−p) p^2 . The name “inverse binomial” would perhaps be more a ...

332 CHAPTER 6 Inferential Statistics E(X) = E(X 1 +X 2 +X 3 ) = E(X 1 )+E(X 2 )+E(X 3 ) = 1+ 3 2 +3 = 11 2 . The generalization ...

SECTION 6.1 Discrete Random Variables 333 In order to appreciate the method em- ployed, let’s again consider the geo- metric dis ...

334 CHAPTER 6 Inferential Statistics • HH HH HHT (B 1 = 0 and we start the experiment over again with one trial already having b ...

SECTION 6.1 Discrete Random Variables 335 E(X) = max∑{n,k} m=0 m Än m äÄN−n k−m ä ÄN k ä. We can calculate the above using simpl ...

336 CHAPTER 6 Inferential Statistics x^2  d 2 dx^2 (x+ 1)n  (x+ 1)N−n = n(n−1)x^2 (x+ 1)N−^2 = x^2 n(n−1) N(N−1) d^2 dx^2 (x ...

SECTION 6.1 Discrete Random Variables 337 Var(X) = E(X^2 )−E(X)^2 = n(n−1)k(k−1) N(N−1) + nk N − (nk N ) 2 = nk(N−n)(N−k) N^2 (N ...

338 CHAPTER 6 Inferential Statistics Next, note that the limit above is a 1∞indeterminate form; taking the natural log and apply ...

SECTION 6.1 Discrete Random Variables 339 We expect that the mean of the Poisson random variable isμ; how- ever, a direct proof ...

340 CHAPTER 6 Inferential Statistics Exercises Suppose that you are going to toss a fair coin 200 times. Therefore, you know th ...

SECTION 6.1 Discrete Random Variables 341 My motorcycle has a really lousy starter; under normal conditions my motorcycle will ...

342 CHAPTER 6 Inferential Statistics xk), where x 1 +x 2 +···+xk =n. A little thought reveals that these probabilities are given ...

SECTION 6.1 Discrete Random Variables 343 (c) That is, ifAstarts witha dollars andbstarts withbdollars, then the probability tha ...

344 CHAPTER 6 Inferential Statistics (d) the probability that you selected at most 2 white balls (using a binomial model). LetX ...

SECTION 6.1 Discrete Random Variables 345 PROGRAM: PRIZES :Input “NO OF PRIZES: ”, M :Input “NO OF TRIALS: ”, N :0→A :For(I,1,N) ...

346 CHAPTER 6 Inferential Statistics thatE must be infinite. (b) Here we’ll give a nuts and bolts direct approach.^10 Note first ...

SECTION 6.1 Discrete Random Variables 347 (a) Suppose that we wish to move along a square grid (a 6 × 6 grid is shown to the rig ...

348 CHAPTER 6 Inferential Statistics (b) Show that the maximum probability for a leader to be chosen in a given round occurs whe ...

SECTION 6.2 Continuous Random Variables 349 the meaning of uniformity! What uniformity means is that for any two numbersx 1 andx ...

350 CHAPTER 6 Inferential Statistics 6











1 t y y=fx(t) =2t Two important observations are in order. (a) ...