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# ECON 300 ECON 300

UIC

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This 4 page Class Notes was uploaded by Patricia Soto on Monday January 18, 2016. The Class Notes belongs to ECON 300 at University of Illinois at Chicago taught by Irina Horoi in Winter 2016. Since its upload, it has received 557 views. For similar materials see Econometrics in Economcs at University of Illinois at Chicago.

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Date Created: 01/18/16

ECON 300 Week 1 Notes Random Variable (RV) – A variable whose value is determined by a stochastic process. • Denoted by a capital letter like X. • The value that a random valuable takes on is denoted by a lower case letter like x. • 2 Kinds of RV o Discrete RV – Takes on finite number of values § Ex. The # of successful foul shots made out of 2 attempts = X It can take on any value in the set {0,1,2} § Discrete RV is characterized by its probability density function (PDF) (*Think of PDF as distribution) ú PDF- f(X-subj) = p-subj for subj = 1, 2 ……k ú Here there three possible value of RV X and the jth value occurance is given by p; The pdf free throws makde out of 2 attempts ú All values add to 1 o Continuous RV – Takes on infinite possibilities. Takes on any specific real value with 0 probability. § Ex. Consider the RV age and for simplicity suppose people are evenly distributed between age 0 and 100 years. Let’s measure age in hours instead of years. {0,..100* 365*24 – 876,000} = .00001 (probability of picking any age) § The precision of your measurement dictates how many possible values an RV can take on in reality. § In the real world everything is discrete however, when the # od outcomes with non-zero probability of occurring becomes large you can treat is as continuous. § Continuous RV can be described by a cumulative density function (CDF) ú F(x) = P( X ≤ x) => this funcation takes x + returns the probability that the RV takes on a value less than or equal to x. ú Properties of CDF (Consider constants a, b, c.) • P(x >c) = 1-F(c) • P( a<X≤b) = F(b) – F(a) • P(x≥c) = P(>c) -> useful for hypothesis testing ECON 300 Week 1 Notes o Multiple RVs- when we are concerned with more than one RV we use a joint pdf => f-subx,suby (x,y) = P(x=x, y=y) => This is the probability. o Independence – RVs X and Y are independent if and only if f-subx,suby(x,y) = f-subx(x) * f-suby(y) § f-subx(x) & f-suby(y) are known as marginal probability functions § Just the pdf of own RV that are evaluated at x and y ú Ex: 2 RVs • L: {1,0) (1 = late, 0 = on time) • R: {1,0} (1 = raining, 0 = not raining) Late Not late Raining .10 .10 Not Raining .08 .72 • Are raining and late independent of each other? o Add rows across and down o Late Not Late Raining .10 .10 .20 Not .08 .72 .80 Raining ú .18 .82 ú P-subx*suby = .1 ú .2 ≠ .18 ≠ .36 o Conditional Distribution – The probability of something happening conditional on something else (tells us the probability of x conditional on Y = y § P (Y = y, X = x) = f-suby|subx (y1) = f- subx,suby(x,y)/ f-subx(x) ú | = conditional X = years of schooling Y = income in thousands 30 40 50 F(x) 9 .2 .1 .1 .4 ECON 300 Week 1 Notes 12 .05 .2 .05 .3 16 .1 .00 .2 .3 F(y) .35 .30 .35 P(Y=y|X=x) = f-subysubk = f(x,y)/f(x) = P(Y = 40| X= 12) = .2/.3 = 2/3 *marginal probability = add horizontally E[X] = xsub1 * psub1 + xsub2* psub2 + …..xsubn * psubn Ex. Rolling Dice • E[Roll] = 1/6[ 1+2+3+4+5+6] E[Y] = 30*.035 + 40*.03 +50*.05 =40 Properties of Expectations E[c] = c (distribution = 1 at whatever the value is) E[aX + b] = E[aX] + E[b] (E[b] = b (because of first property)) =aE[x] +b E[X +Y] = E[X] + E[Y] Median – What is the middle value of the distribution, used for non- symmetrical distributions (i.e very skewed) • Formal definition: The middle value. A random draw from the distribution will have an equal value of being above the median as being below the median. F(x-subm) = .5 F = cdf x-subm =median Right tailed distribution : median > mean Conditional Expectations – Just like expectations but with given values E[Y|X=x] =Σ y-sub0 * f-suby|x(y-sub0|x) Properties of Conidtional Expectations 1. E[x(x)|X] = c(x) 2. E[a(x)*Y +b(x)|X] = a(x)E[Y|X] + b(x) 3. If X and Y are independent, E [Y|X] = E[Y] 4. The Law of Iterated Expectations • E-suby[E[Y|X]] =E[Y] Variances Var(x) = E[(x-µ)^2] = E[x^2] - µ^2 = σ^2 Properties of Variance 1. Var(c) = o 2. Var(aX +c) = Var(aX + Var(b) = a^2Var(X) If X and Y are independent: 3. Var(X+Y) = Var(X) + Var(Y) ECON 300 Week 1 Notes 4. Var(Y-X) = Var(X) + Var(Y) Standard Deviation – Variance^2 Covariance – measure of how two RV move with each other Cor(X,Y) = E[(X-µsubx)(Y-µsuby)]= σsubxsuby Properties 1. Cov(X,Y) =0 if X,Y are independent 2. Cov(aX +b, cY+d) = a*c*Cov(X,Y) 3. |Cov(X,Y)| ≤ sd(X) *sd(Y) Correlation – measure of linear strength Corr(X,Y) = Cov(X,Y)/sd(x)*sd(Y) (varies from -1 to 1)

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