Handout Ch 4 實習

微積分複習第二波(1) Example Jia-Ying Chen

微積分複習第二波(2) 變數變換 Example 這是什麼鬼 Jia-Ying Chen

微積分複習第二波(3) Ch 4積分補充 Jia-Ying Chen

Find x Jia-Ying Chen

新的消去法? Jia-Ying Chen

Another way to expand an equation Jia-Ying Chen

歸去來析 ( 乾脆去死,台語) Jia-Ying Chen

Expectation of a Random Variable Discrete distribution Continuous distribution E(X) is called expected value, mean or expectation of X. E(X) can be regarded as being the center of gravity of that distribution. E(X) exists if and only if E(X) exists if and only if Whenever X is a bounded random variable, then E(X) must exist. Jia-Ying Chen

The Expectation of a Function Let , then Let , then Suppose X has p.d.f as follows: Let it can be shown that Jia-Ying Chen

Example 1 (4.1.3) In a class of 50 students, the number of students ni of each age i is shown in the following table: If a student is to be selected at random from the class, what is the expected value of his age Agei 18 19 20 21 25 ni 22 4 3 1 Jia-Ying Chen

Solution E[X]=18*0.4+19*0.44+20*0.08+21*0.06+ 25*0.02=18.92 Agei 18 19 ni 22 4 3 1 Pi 0.4 0.44 0.08 0.06 0.02 Jia-Ying Chen

Properties of Expectations If there exists a constant such that If are n random variables such that each exists, then For all constants Usually Only linear functions g satisfy If are n independent random variable such that each exists, then Jia-Ying Chen

Example 2 (4.2.6) Suppose that a particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump, the probability is p (0<=p<=1) that the particle will jump one unit to the left and the probability is 1-p that the particle will jump one unit to the right. Find the expected value of the position of the particle after n jumps. Jia-Ying Chen

Solution Jia-Ying Chen

Properties of the Variance Var(X ) = 0 if and only if there exists a constant c such that Pr(X = c) = 1. For constant a and b, . Proof : Jia-Ying Chen

Properties of the Variance If X1 , …, Xn are independent random variables, then If X1,…, Xn are independent random variables, then Jia-Ying Chen

Example 3 (4.3.6) Suppose that X and Y are independent random variables with finite variances such that E(X)=E(Y) Show that Jia-Ying Chen

Solution Jia-Ying Chen

Moment Generating Functions Consider a given random variable X and for each real number t, we shall let . The function is called the moment generating function (m.g.f.) of X. Suppose that the m.g.f. of X exists for all values of t in some open interval around t = 0. Then, More generally, Jia-Ying Chen

Properties of Moment Generating Functions Let X has m.g.f. ; let Y = aX+b has m.g.f. . Then for every value of t such that exists, Proof: Suppose that X1,…, Xn are n independent random variables; and for i = 1,…, n, let denote the m.g.f. of Xi. Let , and let the m.g.f. of Y be denoted by . Then for every value of t such that exists, we have Jia-Ying Chen

The m.g.f. for the Binomial Distribution Suppose that a random variable X has a binomial distribution with parameters n and p. We can represent X as the sum of n independent random variables X1,…, Xn. Determine the m.g.f. of Jia-Ying Chen

Uniqueness of Moment Generating Functions If the m.g.f. of two random variables X1 and X2 are identical for all values of t in an open interval around t = 0, then the probability distributions of X1 and X2 must be identical. The additive property of the binomial distribution Suppose X1 and X2 are independent random variables. They have binomial distributions with parameters n1 and p and n2 and p. Let the m.g.f. of X1 + X2 be denoted by . The distribution of X1 + X2 must be binomial distribution with parameters n1 + n2 and p. Jia-Ying Chen

Example 4 (4.4.10) Suppose that the random variables X and Y are i.i.d. and that the m.g.f. of each is Find the m.g.f. of Z=2X-3Y+4 Jia-Ying Chen

Solution Jia-Ying Chen

Properties of Variance and Covariance If X and Y are random variables such that and , then Correlation only measures linear relationship. 兩各變數可以dependent,但correlation=0 Example: Suppose that X can take only three values –1, 0, and 1, and that each of these three values has the same probability. Let Y=X 2. So X and Y are dependent. E(XY)=E(X 3)=E(X)=0, so Cov(X,Y) = E(XY) – E(X)E(Y)=0 (uncorrelated). Jia-Ying Chen

Example 5 (4.6.12) Suppose that X and Y have a continuous joint distribution for which the joint p.d.f is as follows: Determine the value of Var(2X-3Y+8) Jia-Ying Chen

Solution Jia-Ying Chen