Advanced High-School Mathematics

SECTION 6.2 Continuous Random Variables 351 In this case thenormalrandom variableXcan assume any real value. Furthermore, it is ...

352 CHAPTER 6 Inferential Statistics does the underlying density curve look like? Is it still uniform as in the case ofX? Probab ...

SECTION 6.2 Continuous Random Variables 353 differentiating both sides with respect totand applying the Fundamen- tal Theorem of ...

354 CHAPTER 6 Inferential Statistics (b) EstimateP(. 5 ≤Z ≤ 1 .65) through a simulation, using the TI code as follows. (I would ...

SECTION 6.2 Continuous Random Variables 355 (c) the probability that the quadraticx^2 +Bx+C = 0 has two real roots. Do the same ...

356 CHAPTER 6 Inferential Statistics (b) Write a TI program to generate 200 samples ofY. (c) Graph the histogram generated in (b ...

SECTION 6.2 Continuous Random Variables 357 6 x y χ^2 distribution with four degrees of freedom χ^2 distribution with one de ...

358 CHAPTER 6 Inferential Statistics circles intersect. (Hint: letX 1 andX 2 be independent instances of randand note that it su ...

SECTION 6.2 Continuous Random Variables 359 onk. This says that during the game we don’t “age”; our probability of dying at the ...

360 CHAPTER 6 Inferential Statistics F′(t+τ)F(t) =F(t+τ)F′(t). But this can be written as d dt lnF(t+τ) = d dt lnF(t), forcing F ...

SECTION 6.2 Continuous Random Variables 361 number of days between accidents at a given intersection. Exercises Prove the ass ...

362 CHAPTER 6 Inferential Statistics You have determined that along a stretch of highway, you see on average one dead animal on ...

SECTION 6.2 Continuous Random Variables 363 of a geometric random variable. Just as we were able above to transformrandinto an e ...

364 CHAPTER 6 Inferential Statistics 1 −(1−p)k = P(Y ≤k) = P(g(X)≤k) = P(X≤g−^1 (k)) = g−^1 (k) n Solving forgwe get g(h) = ln(1 ...

SECTION 6.3 Parameters and Statistics 365 LetXbe an exponential random variable with failure rateλ, and letY =X^1 /α, α >0. ...

366 CHAPTER 6 Inferential Statistics Var(X) = ∫∞ −∞(x−μX) (^2) fX(x)dx. (though most texts gloss over this point). The positive ...

SECTION 6.3 Parameters and Statistics 367 E(X 1 +X 2 +···+Xk) =E(X 1 ) +E(X 2 ) +···+E(Xk). If the random variablesXandY are ind ...

368 CHAPTER 6 Inferential Statistics ∫t−a −∞ fX(x)dx = ∫t −∞fX(x−a)dx. In other words, for all real numberst, we have ∫t −∞fY(x) ...

SECTION 6.3 Parameters and Statistics 369 This proves the assertion made on page 365. Next, we have E(X^2 ) = ∫∞ 0 xfX (^2) (x)d ...

370 CHAPTER 6 Inferential Statistics Convolution and the sum of independent random variables. Assume that XandY are independent ...