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# cheatsheet140 - Chapter 2: normal distribution formula:...

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Chapter 2: normal distribution formula: f(x) = 1/ sqrt(2∏)exp(-x 2 /2) Binomial distribution: f(x) = (n/x)p x q (n-x) = (n!/(x!(n- x)!)p x q n-x Mean of a distribution = μ = E(x) = ∑[x i f(x i )] E[g(x)] = ∑[g(x i )f(x i )] Property 2.1: a. E(x-μ) = E(x) – μ = 0 b. If c is a constant or is nonrandom variable, E(c) = c c. If c is a constant or is nonrandom, E[cg(X)] = cE(g(X)] d. E[u(x) + v(x)] = E[u(x)] + E[v(x)] Variance of X is defined as: σ 2 = var(X) = E[(X – μ) 2 ] = ∑(x i – μ) 2 f(x i ) Properties: a. σ 2 = E[(X-μ) 2 ] = E[X 2 - 2μX + μ 2 ] = E(x 2 ) - 2μE(X) + μ 2 = E(X 2 ) – μ 2 b. It follows from this that if c is a constant or is nonrandom, Var(c) = 0 c. If a and b are constants or nonrandom , Var(a+bX) = b 2 σ 2 Normal Distribution properties: a. It is symmetric around the mean value μ and has a bell shape b. The area between 1S.D. is 68.26% The area between 2S.D. is 95.44% and between 3 S.d. it is 99.73% c. The standardized random variable Z = (x – μ)/σ has the distribution N(0,1) d. If X is distributed as N(μ, σ 2 ) then Y = a +bX, where a and b are fixed constants, is distributed N(a + bμ, b 2 σ 2 ) E([a(X) + b(X)Y] | X) = a(X) + b(X)E(Y|X) Cov(X,Y) = E[(X-μx)(Y-μy)] = E(XY – X μy – μ x Y + μ x μ y ] = E(XY) – μ y E(X) = μ x E(Y) + μ x μ y = E(XY) - μ x μ y It readily follows that Cov(X,X) = Var(X) Correlation Coefficient = ρ xy = σ xy / σ x σ y = Cov(X,Y)/ sqrt[Var(X)Var(Y)] Properties: A, If a and b are constants then Var(ax+bx) = a^2var(x)+b^2var(y) +2abcov(x,y) B. correlation coefficient lies between -1 and 1 C. If x’s are independent then cov(xi, xj) = 0 for all Sampling from a Normal Population: X~N(mu, var) a. Z = (x-bar – mu)/(SD/sqrt(n)) Large-Sample Distribution a. The law of large numbers: As n increases, the sample mean of a set of random variables approaches its expected value. b. The central limit theorem: Sampling distribution of Zn=sqrt (n)(x-bar –mu)/SD converges to the standard normal N(0,1) as n converges to infinity. Holds even when the population distribution is not normal Sample covariance: Cov(X,Y) = (1/n-1)sum(x-x-bar) (y-y-bar) Rxy = Sxy/SxSy = cov(X,Y)/SD(X)SD(Y) Method of Least Squares (or Ordinary Least Squares) Most commonly used methods of estimating parameters ESS (mu) = sum(x-mu)\$^2 = sum(x-x-bar)^2 + sum(x- bar-mu)^2 The probability of rejecting a hypothesis when it is false is called the power of a test and is equal to 1- P(Type II error) Interval Estimation: P[x-bar +/– (s/sqrt(n))*t*] = confidence interval (1- significance level) Chapter 3: Y = alpha + beta*X + u (error term) Y-hat = alpha-hat + beta-hat*X The error results because of the following (difference between the Y and estimates alpha and beta): 1. Omitted variables (u thus capture the effect of Z variable) 2. Nonlinearities 3. Measurement errors (errors in measuring X and Y) 4. Unpredictable effects (unpredictable random effects) B – Marginal effect of X on Y (with each increase in

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## This note was uploaded on 04/02/2008 for the course ECON 140 taught by Professor Duncan during the Spring '08 term at Berkeley.

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cheatsheet140 - Chapter 2: normal distribution formula:...

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