Advanced High-School Mathematics

SECTION 5.5 Differential Equations 311 whereM(x,y) andN(x,y) are both homogeneous of thesame degree. These are important since t ...

312 CHAPTER 5 Series and Differential Equations 5.5.3 Linear first-order ODE; integrating factors In this subsection we shall co ...

SECTION 5.5 Differential Equations 313 y x+ 1 = ∫ xdx (x+ 1)^2 = ∫ (x+ 1−1)dx (x+ 1)^2 = ∫ Ñ 1 x+ 1 − 1 (x+ 1)^2 é dx = ln(x+ 1) ...

314 CHAPTER 5 Series and Differential Equations Solve the Bernoulli ODE (a) y′+ 3 x y=x^2 y^2 , x > 0 (b) 2y′+ 1 x+ 1 y+ 2( ...

SECTION 5.5 Differential Equations 315 We can tabulate the results: n xn yn=yn− 1 n xn yn=yn− 1 +F(xn− 1 ,yn− 1 ) +F(xn− 1 ,yn− ...

316 CHAPTER 5 Series and Differential Equations n xn yn=yn− 1 +F(xn− 1 ,yn− 1 )h y(xn) 0 0 1 1 .2 1 2 .4 1.04 3 .6 1.128 4 .8 1. ...

Chapter 6 Inferential Statistics We shall assume that the student has had some previous exposure to elementary probability theor ...

318 CHAPTER 6 Inferential Statistics 6.1 Discrete Random Variables Let’s start with an example which is probably familiar to eve ...

SECTION 6.1 Discrete Random Variables 319 μX = E(X) = ∑ xiP(X=xi), where the sum is over all possible valuesxiwhich the random v ...

320 CHAPTER 6 Inferential Statistics P(Y =y) = ∑∞ j=1 P(Y =y|X=xi)P(X=xi). (6.2) Having noted this, we now proceed: μX+Y = ∑∞ i= ...

SECTION 6.1 Discrete Random Variables 321 Next, we define thevarianceσ^2 (or Var(X)) of the random variable X having mean μ by s ...

322 CHAPTER 6 Inferential Statistics As you might expect, the above formula isfalsein general (i.e., when XandY not independent) ...

SECTION 6.1 Discrete Random Variables 323 Lemma. (Markov’s Inequality)Let Xbe a non-negative discrete ran- dom variable. Then fo ...

324 CHAPTER 6 Inferential Statistics From what we’ve proved about the mean, we see already thatE(X) = μ. In case the random vari ...

SECTION 6.1 Discrete Random Variables 325 Suppose that we draw two cards in succession, and without re- placement, from a stand ...

326 CHAPTER 6 Inferential Statistics Eric, who then tosses the coin. If the result is heads, Eric wins; otherwise he returns the ...

SECTION 6.1 Discrete Random Variables 327 We can give an intuitive idea of how one can analyze this question, as follows. We sta ...

328 CHAPTER 6 Inferential Statistics E(X) = ∑∞ n=1 nP(X=n) = ∑∞ n=1 np(1−p)n−^1 = p ∑∞ n=1 n(1−p)n−^1. Note that ∑∞ n=1 n(1−p)n− ...

SECTION 6.1 Discrete Random Variables 329 Next, ∑∞ n=1 n^2 (1−p)n−^1 = ∑∞ n=1 n(n−1)(1−p)n−^1 + ∑∞ n=1 n(1−p)n−^1 = (1−p) ∑∞ n=1 ...

330 CHAPTER 6 Inferential Statistics Next, ifX is the binomial random variable with success probabilityp, then we may write X = ...